Triangles

Problem 1

In triangle $D E F$ , find the longest side given $m \angle D=59^{\circ}, m \angle E=76^{\circ}, m \angle F=45^{\circ}$ .

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Problem 2

Which type of triangle cannot be both obtuse and one of the following: equilateral, acute, isosceles, or scalene?

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Problem 3

Find the side lengths of $\triangle ABC$ with ratio $2:3:4$ and perimeter 108.

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Problem 4

Which type of triangle cannot be both scalene and acute, right and acute, equilateral and acute, or isosceles and acute?

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Problem 5

Find the roller reaction at point A of a symmetric fink roof truss with $s = 3m$ , $h = 1.5m$ , and a vertical load $P = 25kN$ .

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Problem 6

Find the sum of the perimeters of the four triangles formed by the diagonals of square $A B C D$ . Choices: A. $2+2 \sqrt{2}$ B. $8+4 \sqrt{2}$ C. $8+8 \sqrt{2}$ D. 16

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Problem 7

Find the height $h$ of triangle $ABC$ with area $27 \mathrm{~cm}^2$ and base $AB=6 \mathrm{~cm}$ .

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Problem 8

Find the largest perimeter of an isosceles triangle with sides $8.2 \mathrm{~cm}$ and $9.4 \mathrm{~cm}$ measured to the nearest mm.

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Problem 9

Le mât mesure 3,6 m. Vérifie si les triangles VEN et VNT sont rectangles avec les longueurs 4,2 m et 3,9 m. Justifie.

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Problem 10

Find the area of a triangle with sides 3, 4, and 5. Options: a) 6 b) 7.5 c) 10 d) 12 e) 15.

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Problem 11

If a triangle is scalene, then all side lengths are different. Choose the equivalent statement: A, B, or C.

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Problem 12

Determine which sets can be side lengths of a triangle: A. $9,5,15$ B. $10,3,7$ C. $8,5,7$

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Problem 13

Find side $BC$ of triangle $ABC$ with $AB=8$ , $AC=8\sqrt{2}$ , $\angle ABC=45^{\circ}$ , $\angle ACB=30^{\circ}$ .

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Problem 14

In triangle $ABC$ , with $BC=3\sqrt{2}$ and $\angle ABC=45^{\circ}$ , find the minimum of $CM + MN$ as $M$ and $N$ move on $BD$ and $BC$ .

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Problem 15

Identify valid triangle side lengths from these options: A. $16,8,10$ B. $4,12,6$ C. $6,9,17$

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Problem 16

If the sides of an equilateral triangle are reduced by $2 \mathrm{~cm}$ , the new perimeter is $\frac{4}{5}$ of the original. Find the original side length.

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Problem 17

Sketch two isosceles triangles with one angle $40^{\circ}$ , marking all vertices and angles. Remember, angles sum to $180^{\circ}$ .

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Problem 18

Divide a square $ABCD$ with side length $4 \mathrm{~cm}$ into 4 equal triangles using 2 line segments. Explain how.

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Problem 19

Find the value of $x$ in a triangle with angles $x, 43^{\circ}, 79^{\circ}$ .

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Problem 20

Demonstrați că în triunghiul dreptunghic, lungimea medianei pe ipotenuză este $A O=\frac{B C}{2}$ .

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Problem 21

Find the perimeter of triangle ABC where $AF=6$ , $CD=5$ , and $BE=4$ .

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Problem 22

Un voilier a un mât de $3,6 \mathrm{~m}$ avec un étai de $4,2 \mathrm{~m}$ et des haubans de $3,9 \mathrm{~m}$ . Est-ce que les triangles VEN et VNT sont rectangles ? Justifie.

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Problem 23

Find the length of the third side of a triangle with sides 30.3 m, 18.5 m, and an angle of $61^{\circ}$ between them.

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Problem 24

Find the third side of a triangle with sides 30.3 m and 18.5 m, and an angle of $61^{\circ}$ . Options: 24.1 m, 26.8 m, 30.0 m, 32.6 m.

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Problem 25

Find the length of the third side of a triangle with sides $600 \mathrm{~m}$ , $250 \mathrm{~m}$ , and opposite angle $34^{\circ}$ .

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Problem 26

Find the length of the third side of a triangle with sides $45.3 \mathrm{~m}$ , $29.0 \mathrm{~m}$ , and angle $26^{\circ}$ .

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Problem 27

Find the area of triangle formed by points A(3,5), B(7,5), and C(7,-1).

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Problem 28

In a coordinate plane, $\triangle ABC$ is a right triangle with $A(-4,1)$ and $B(2,1)$ . Area is 9 sq. units. Find point $C$ .

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Problem 29

In a coordinate plane, $\triangle ABC$ has $A(-4,1)$ and $B(2,1)$ with area 9 sq. units. Find other $C$ coordinates besides $(2,4)$ .

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Problem 30

In triangle ABC, DE and FG are parallel to CB. Given AE = 9, EG = 8, GB = 7, find the length of AC in inches.

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Problem 31

Triangle $ABC$ has an area of $108 \, \text{cm}^2$ . If base $x$ and height $y$ are integers, find possible values for $x$ . Choices: (A) 4, (B) 5, (C) 6, (D) 8, (E) 9.

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Problem 32

In a triangle, if side $x = \frac{3}{5} z$ and $y = \frac{4}{3} x$ , find the ratio of $y$ to $z$ .

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Problem 33

A right triangle has an area of 16 square miles. One leg is 4 miles longer than the other. Find the leg lengths. $b(b+4)=32$ $\frac{1}{2} \times b \times h=16$

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Problem 34

A right triangle has an area of 16 square miles. One leg is 4 miles longer than the other. Find the leg lengths.

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Problem 35

Find the height of a scalene triangle with base $b = 4$ units and area $A = 16$ square units.

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Problem 36

Find the base of a triangle with height $12 \mathrm{~cm}$ more than base and area $32 \mathrm{~cm}^{2}$ . Options: A. $4 \mathrm{~cm}$ B. $8 \mathrm{~cm}$ C. $12 \mathrm{~cm}$ D. $16 \mathrm{~cm}$

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Problem 37

Three charges of $+6.65 \mu \mathrm{C}$ at the triangle's vertices create a force. Find the resultant force: a. $172.34 \mathrm{~N}$ b. $22.98 \mathrm{~N}$ c. $34.47 \mathrm{~N}$ d. $68.94 \mathrm{~N}$ e. $0.89 \mathrm{~N}$

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Problem 38

Find angle $B$ in triangle $ABC$ if $A=15^{\circ}$ and $C=50^{\circ}$ .

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Problem 39

Find the perimeter of triangle PQR with vertices at $P(-10,-2)$ , $Q(-10,6)$ , and $R(5,2)$ .

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Problem 40

Find the perimeter of triangle PQR with vertices at $P(-10,-2)$ , $Q(-10,6)$ , and $R(5,2)$ .

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Problem 41

Find the area of the triangle with vertices $(4,4)$ , $(4,0)$ , and $(0,0)$ by counting whole and half squares. 8 sq. units. Which quadrant is the point $(-3,2)$ in? Second.

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Problem 42

Prove that $c^{2}=a^{2}+b^{2}-2ab\cos C$ .

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Problem 43

Triunghiul echilateral $ABC$ are perimetrul $27 \mathrm{~cm}$ . Găsește lungimea segmentului $MC$ , unde $AN=6 \mathrm{~cm}$ .

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Problem 44

Найдите периметр равнобедренного треугольника, если одна сторона $a = 7$ дм, а другая $b = a + 8$ дм.

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Problem 45

Una gaviota observa un bote a $42^{\circ}$ de depresión desde 45 m de altura. ¿Cuál es la distancia horizontal al barco?

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Problem 46

Desde una torre de 28 m, un salvavidas ve a una persona con un ángulo de depresión de $58^{\circ}$ . ¿Cuál es la distancia a la persona?

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Problem 47

已知圆O的半径为2.5，弦AB的长度为2，若三角形ABC为等腰三角形，则求 $BC^{2}=$ 。

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Problem 48

Calculați aria triunghiului $B O D$ și măsura unghiului $B A D$ în patrulaterul convex $A B C D$ cu datele date.

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Problem 49

4. Într-un pătrat $A B C D$ , cu punctele $E, F, G$ pe laturi, arată că triunghiul $D E G$ este isoscel și află unghiul dintre $D F$ și $E G$ .

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Problem 50

5. Într-un dreptunghi cu $AB=12 \mathrm{~cm}$ și $BC=4 \mathrm{~cm}$ , calculează aria triunghiului $CEO$ și unghiul $BFC$ .

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Problem 51

Un topógrafo mide el ángulo de elevación al cerro: $45^{\circ}$ y a $800$ m más, $30^{\circ}$ . ¿Cuál es la altura del cerro?

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Problem 52

Calculate the distance $BC$ in triangle $ABC$ where $AB=1.5 \text{ km}$ , $AC=1.2 \text{ km}$ , and $\angle A=50^\circ$ .

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Problem 53

等腰三角形一个底角为 45 度，问这个三角形的类型是？(1) 锐角 (2) 直角 (3) 钝角 (4) 无法确定

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Problem 54

1. Într-o livadă sunt 160 de pomi. În prima zi s-au plantat $60\%$ meri și $40\%$ nuci, iar în a doua zi 76 pomi. Află procentul din prima zi și câți meri și nuci s-au plantat.
2. Consideră $E(x)=x^{2}-(x+1)^{2}-(x+2)^{2}+(x+3)^{2}$ . Arată că $E(x)=4$ pentru orice $x$ și calculează suma $S=1^{2}-2^{2}-3^{2}+4^{2}+\ldots+40^{2}$ .
3. În sistemul de axe $x O y$ , $A(3,2)$ , $B$ simetricul lui $A$ față de $O x$ , și $C$ simetricul lui $A$ față de $O$ . Arată că $C=(-3,-2)$ și determină funcția corespunzătoare lui $BC$ .
4. Semicercul de centru $D$ este tangent la laturile $A B$ și $A C$ ale triunghiului isoscel $A B C$ . Demonstrează că $D$ este mijlocul lui $B C$ și află raza semicercului, știind că $B C=30 \mathrm{~cm}$ și $A D=20 \mathrm{~cm}$ .

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Problem 55

Find the perimeter of triangle $ABC$ with $AD=4 \sqrt{3} cm$ , $BC=5 cm$ , and $BD=1 cm$ .

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Problem 56

Find the expression for the area of a triangular attic floor with height $n \mathrm{yd}$ and base $6 \mathrm{yd}$ .

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Problem 57

삼각형의 변이 $x-1, x, x+1$ 일 때, $x$ 의 가능한 값 중 아닌 것은? (1) 2 (2) 3 (3) 4 (4) 5 (5) 6

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Problem 58

Calcola perimetro e area di un triangolo rettangolo con cateti di $10 \mathrm{~cm}$ e $24 \mathrm{~cm}$ .

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Problem 59

Calcola il perimetro e l'area di un triangolo rettangolo con ipotenusa $75 \mathrm{~cm}$ e cateto $45 \mathrm{~cm}$ .

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Problem 60

Calcola perimetro e area di un triangolo rettangolo con ipotenusa $75 \mathrm{~cm}$ e cateto $45 \mathrm{~cm}$ .

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Problem 61

Un triangolo rettangolo ha area $180 \mathrm{~cm}^{2}$ e un cateto di $9 \mathrm{~cm}$ . Trova il perimetro.

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Problem 62

Berechne die fehlende Kathetenlänge $b$ eines rechtwinkligen Dreiecks mit Fläche $A=490mm^2$ und Kathete $a=28mm$ .

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Problem 63

Gegeben sind $A=5 \, \text{cm}^{2}$ , $a=80 \, \text{mm}$ und $\gamma=90^{\circ}$ . Welche geometrische Figur ist gefragt?

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Problem 64

Berechne die fehlende Kathetenlänge eines rechtwinkligen Dreiecks mit $\gamma=90^{\circ}$ , Fläche 5 cm $^2$ und Kathete 80 mm.

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Problem 65

Bài 1. Cho tam giác vuông $\mathrm{ABC}$ với $\mathrm{AB}=30 \mathrm{~cm}$ , cao $\mathrm{AH}=24 \mathrm{cm}$ . Tính $\mathrm{BH}, \mathrm{BC}, \mathrm{AC}$ và độ dài $\mathrm{HD}$ với $\mathrm{HD} \perp \mathrm{AC}$ .

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Problem 66

Un triangolo isoscele ha base $32 \mathrm{~cm}$ e altezza $12 \mathrm{~cm}$ . Trova perimetro e area.

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Problem 67

Calcola l'area di un triangolo isoscele con perimetro $36 \mathrm{~cm}$ e lato obliquo $13 \mathrm{~cm}$ .

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Problem 68

Diberi $AB = AC$ dan kecerunan $AB = \frac{2}{3}$ . Jika $BC = 12$ cm, cari tinggi $A$ dari $BC$ .

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Problem 69

Find the area of a triangle with a chord of length $10 \mathrm{~cm}$ in a circle of radius $13 \mathrm{~cm}$ .

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Problem 70

In un triangolo rettangolo, i cateti sommano a $21 \mathrm{~cm}$ e differiscono di $3 \mathrm{~cm}$ . Trova l'altezza sull'ipotenusa di $15 \mathrm{~cm}$ .

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Problem 71

In un triangolo rettangolo, i cateti sommano a $21 \mathrm{~cm}$ e differiscono di $3 \mathrm{~cm}$ . Trova l'altezza sull'ipotenusa di $15 \mathrm{~cm}$ .

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Problem 72

Calcola il perimetro di un triangolo isoscele con area di $672 \mathrm{~cm}^{2}$ e base di $28 \mathrm{~cm}$ .

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Problem 73

Un triangolo ha area $14,95 \mathrm{~cm}^{2}$ e un cateto di $6,5 \mathrm{~cm}$ . Trova l'altro cateto e l'altezza sull'ipotenusa di $13 \mathrm{~cm}$ .

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Problem 74

In un triangolo rettangolo, un cateto è $36 \mathrm{~cm}$ e l'altro è $\frac{4}{3}$ volte il primo. Trova perimetro e area.

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Problem 75

Bài 1. Cho tam giác $ABC$ vuông tại $A$ với $\mathrm{AB}=30 \mathrm{~cm}$ , cao $\mathrm{AH}=24 \mathrm{cm}$ . Tính $BH$ , $BC$ , $AC$ và $HD$ .

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Problem 76

A triangle has sides of 6 in and 26 in. Find the area of a similar triangle with sides half the length of the original.

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Problem 77

Calculate the distance $BC$ in triangle $ABC$ where $AB=1.5$ km, $AC=1.2$ km, and $\hat{A}=50^{\circ}$ .

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Problem 78

In un triangolo rettangolo, i cateti sommano a $21 \mathrm{~cm}$ e differiscono di $3 \mathrm{~cm}$ . Trova l'altezza sull'ipotenusa di $15 \mathrm{~cm}$ .

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Problem 79

Calculate the distance $BC$ in triangle $ABC$ where $AB=1.5$ km, $AC=1.2$ km, and $\hat{A}=50^\circ$ .

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Problem 80

A surveyor sees a building at $34^{\circ}$ elevation, 2544 ft away. Find distance to the building, building height, antenna angle, and height.

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Problem 81

Berechne den Umfang $u$ und den Flächeninhalt $A$ des Dreiecks mit $a=8cm$ , $b=5.66cm$ , $c=8cm$ , $h_c=5.3cm$ .

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Problem 82

Berechne die Höhe $h_a$ eines Dreiecks mit Flächeninhalt $A = 17,55 \, \text{cm}^2$ und Seite $a = 7,8 \, \text{cm}$ .

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Problem 83

Triangle $\mathrm{ABC}$ a $\mathrm{AB}=6 \mathrm{~cm}$ , $\mathrm{BC}=3 \mathrm{~cm}$ , $\mathrm{AC}=4 \mathrm{~cm}$ . Trouvez M, Q, et analysez le triangle $MAB$ .

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Problem 84

In un triangolo rettangolo, l'ipotenusa è lunga $2 \mathrm{~cm}$ più di un cateto e la somma dei cateti è $32 \mathrm{~cm}$ . Trova il perimetro.

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Problem 85

Check if the scale from triangle with sides 1cm, 2cm, 2.5cm to triangle with sides 3cm, 6cm is $1:3$ .

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Problem 86

Find $BC$ if $BC=x-8$ , $AC=2x-8$ , and $AB=12$ .

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Problem 87

Berechne den Flächeninhalt $A$ des Dreiecks und die gesuchten Längen: a) $h_a$ , b) $h_b$ , c) $h_c$ , d) $h_b$ .

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Problem 88

Berechne den Flächeninhalt $A$ des Dreiecks und die gesuchten Längen: a) $c=7,2 \mathrm{~cm}, a=4 \mathrm{~cm}, h_{c}=3,5 \mathrm{~cm} ; h_{a}=$ ? b) $a=9,6 \mathrm{~cm}, b=4 \mathrm{~cm}, h_{a}=5,5 \mathrm{~cm} ; h_{b}=$ ? c) $b=12 \mathrm{~cm}, h_{b}=9,5 \mathrm{~cm}, h_{c}=19 \mathrm{~cm} ; c=$ ? d) $c=11,2 \mathrm{~cm}, h_{c}=7,7 \mathrm{~cm}, h_{b}=5,6 \mathrm{~cm} ; b=$ ?

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Problem 89

Berechne die gesuchten Längen für die gegebenen Flächeninhalte und Höhen der Dreiecke:
a) $A=17,55 \mathrm{~cm}^{2}, a=7,8 \mathrm{~cm}, h_{a}=$ ?
b) $A=2340 \mathrm{~mm}^{2}, h_{b}=36 \mathrm{~mm}, b=$ ?
c) $A=18,25 \mathrm{~cm}^{2}, c=73 \mathrm{~mm}, h_{c}=$ ?
d) $A=34,1 \mathrm{~cm}^{2}, h_{a}=62 \mathrm{~mm}, a=$ ?

See Solution

Problem 90

Berechne die Höhe auf die Hypotenuse $c$ für die gegebenen Seitenlängen: a) $c=5 \mathrm{~cm}, a=3 \mathrm{~cm}, b=4 \mathrm{~cm}$ ; b) $c=25 \mathrm{~cm}, a=15 \mathrm{~cm}, b=20 \mathrm{~cm}$ .

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Problem 91

Measure Sandy's tree height over the years. Draw a slope triangle and check if it shows constant growth. Is the growth linear?

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Problem 92

Sandy measures her tree's height each year: (0,0), (1,1), (2,2), (3,2.5), (4,2.5). Analyze slope triangles to check growth rate.

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Problem 93

Ethan delivers newspapers. Use points (5,200) and (10,400) to find the slope, draw a triangle, and explain its meaning.

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Problem 94

A triangular block has faces at $45^{\circ}$ and $30^{\circ}$ . Find speed of $A$ after both particles travel $0.4 \mathrm{~m}$ .

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Problem 95

Is it true that in an isosceles right triangle, the hypotenuse is $\sqrt{2}$ times one leg's length? A. Yes B. No C. Maybe D. Sometimes E. Not applicable

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Problem 96

If one side's square equals the sum of the other two sides' squares, the triangle is a(n) ___? A. Pythagorean B. acute C. right D. isosceles E. similar

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Problem 97

In a 30-60 right triangle, if the hypotenuse is $x$ times the short leg and the long leg is $y$ times the short leg, find $x$ and $y$ . A. $\sqrt{3}, 2$ B. $2 ; \sqrt{3}$ C. $2 ; \sqrt{2}$ D. $3 ; \sqrt{2}$ E. $\sqrt{2} ; \sqrt{3}$

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Problem 98

Find the length of a slide at a $35^{\circ}$ angle with an $8$ ft base. Estimate to two decimal places. Choices: A. $3.24$ ft B. $16.64$ ft C. $6.06$ ft D. $9.77$ ft E. None.

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Problem 99

If $\triangle ABC \sim \triangle XYZ$ , complete this proportion: $\frac{XY}{YZ}=\frac{?}{?}$ A. $\frac{BC}{CB}$ B. $\frac{AB}{BC}$ C. $\frac{BC}{AB}$ D. $\frac{YZ}{YX}$ E. None of the above

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Problem 100

Triangles $\triangle ABC$ and $\triangle XYZ$ are similar with a ratio of $8:3$ . If $XY=9$ , find $YZ$ .

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Triangles

Problem 1

In triangle DEF D E F DEF, find the longest side given m∠D=59∘,m∠E=76∘,m∠F=45∘ m \angle D=59^{\circ}, m \angle E=76^{\circ}, m \angle F=45^{\circ} m∠D=59∘,m∠E=76∘,m∠F=45∘.

Problem 2

Which type of triangle cannot be both obtuse and one of the following: equilateral, acute, isosceles, or scalene?

Problem 3

Find the side lengths of △ABC \triangle ABC △ABC with ratio 2:3:4 2:3:4 2:3:4 and perimeter 108.

Problem 4

Which type of triangle cannot be both scalene and acute, right and acute, equilateral and acute, or isosceles and acute?

Problem 5

Find the roller reaction at point A of a symmetric fink roof truss with s=3ms = 3ms=3m, h=1.5mh = 1.5mh=1.5m, and a vertical load P=25kNP = 25kNP=25kN.

Problem 6

Find the sum of the perimeters of the four triangles formed by the diagonals of square ABCDA B C DABCD. Choices: A. 2+222+2 \sqrt{2}2+22​ B. 8+428+4 \sqrt{2}8+42​ C. 8+828+8 \sqrt{2}8+82​ D. 16

Problem 7

Find the height hhh of triangle ABCABCABC with area 27 cm227 \mathrm{~cm}^227 cm2 and base AB=6 cmAB=6 \mathrm{~cm}AB=6 cm.

Problem 8

Find the largest perimeter of an isosceles triangle with sides 8.2 cm8.2 \mathrm{~cm}8.2 cm and 9.4 cm9.4 \mathrm{~cm}9.4 cm measured to the nearest mm.

Problem 9

Le mât mesure 3,6 m. Vérifie si les triangles VEN et VNT sont rectangles avec les longueurs 4,2 m et 3,9 m. Justifie.

Problem 10

Find the area of a triangle with sides 3, 4, and 5. Options: a) 6 b) 7.5 c) 10 d) 12 e) 15.

Problem 11

If a triangle is scalene, then all side lengths are different. Choose the equivalent statement: A, B, or C.

Problem 12

Determine which sets can be side lengths of a triangle: A. 9,5,159,5,159,5,15 B. 10,3,710,3,710,3,7 C. 8,5,78,5,78,5,7

Problem 13

Find side BCBCBC of triangle ABCABCABC with AB=8AB=8AB=8, AC=82AC=8\sqrt{2}AC=82​, ∠ABC=45∘\angle ABC=45^{\circ}∠ABC=45∘, ∠ACB=30∘\angle ACB=30^{\circ}∠ACB=30∘.

Problem 14

In triangle ABCABCABC, with BC=32BC=3\sqrt{2}BC=32​ and ∠ABC=45∘\angle ABC=45^{\circ}∠ABC=45∘, find the minimum of CM+MNCM + MNCM+MN as MMM and NNN move on BDBDBD and BCBCBC.

Problem 15

Identify valid triangle side lengths from these options: A. 16,8,1016,8,1016,8,10 B. 4,12,64,12,64,12,6 C. 6,9,176,9,176,9,17

Problem 16

If the sides of an equilateral triangle are reduced by 2 cm2 \mathrm{~cm}2 cm, the new perimeter is 45\frac{4}{5}54​ of the original. Find the original side length.

Problem 17

Sketch two isosceles triangles with one angle 40∘40^{\circ}40∘, marking all vertices and angles. Remember, angles sum to 180∘180^{\circ}180∘.

Problem 18

Divide a square ABCDABCDABCD with side length 4 cm4 \mathrm{~cm}4 cm into 4 equal triangles using 2 line segments. Explain how.

Problem 19

Find the value of xxx in a triangle with angles x,43∘,79∘x, 43^{\circ}, 79^{\circ}x,43∘,79∘.

Problem 20

Demonstrați că în triunghiul dreptunghic, lungimea medianei pe ipotenuză este AO=BC2A O=\frac{B C}{2}AO=2BC​.

Problem 21

Find the perimeter of triangle ABC where AF=6AF=6AF=6, CD=5CD=5CD=5, and BE=4BE=4BE=4.

Problem 22

Un voilier a un mât de 3,6 m3,6 \mathrm{~m}3,6 m avec un étai de 4,2 m4,2 \mathrm{~m}4,2 m et des haubans de 3,9 m3,9 \mathrm{~m}3,9 m. Est-ce que les triangles VEN et VNT sont rectangles ? Justifie.

Problem 23

Find the length of the third side of a triangle with sides 30.3 m, 18.5 m, and an angle of 61∘61^{\circ}61∘ between them.

Problem 24

Find the third side of a triangle with sides 30.3 m and 18.5 m, and an angle of 61∘61^{\circ}61∘. Options: 24.1 m, 26.8 m, 30.0 m, 32.6 m.

Problem 25

Find the length of the third side of a triangle with sides 600 m600 \mathrm{~m}600 m, 250 m250 \mathrm{~m}250 m, and opposite angle 34∘34^{\circ}34∘.

Problem 26

Find the length of the third side of a triangle with sides 45.3 m45.3 \mathrm{~m}45.3 m, 29.0 m29.0 \mathrm{~m}29.0 m, and angle 26∘26^{\circ}26∘.

Problem 27

Find the area of triangle formed by points A(3,5), B(7,5), and C(7,-1).

Problem 28

In a coordinate plane, △ABC\triangle ABC△ABC is a right triangle with A(−4,1)A(-4,1)A(−4,1) and B(2,1)B(2,1)B(2,1). Area is 9 sq. units. Find point CCC.

Problem 29

In a coordinate plane, △ABC\triangle ABC△ABC has A(−4,1)A(-4,1)A(−4,1) and B(2,1)B(2,1)B(2,1) with area 9 sq. units. Find other CCC coordinates besides (2,4)(2,4)(2,4).

Problem 30

In triangle ABC, DE and FG are parallel to CB. Given AE = 9, EG = 8, GB = 7, find the length of AC in inches.

Problem 31

Triangle ABCABCABC has an area of 108 cm2108 \, \text{cm}^2108cm2. If base xxx and height yyy are integers, find possible values for xxx. Choices: (A) 4, (B) 5, (C) 6, (D) 8, (E) 9.

Problem 32

In a triangle, if side x=35zx = \frac{3}{5} zx=53​z and y=43xy = \frac{4}{3} xy=34​x, find the ratio of yyy to zzz.

Problem 33

A right triangle has an area of 16 square miles. One leg is 4 miles longer than the other. Find the leg lengths. b(b+4)=32 b(b+4)=32 b(b+4)=32 12×b×h=16 \frac{1}{2} \times b \times h=16 21​×b×h=16

Problem 34

A right triangle has an area of 16 square miles. One leg is 4 miles longer than the other. Find the leg lengths.

Problem 35

Find the height of a scalene triangle with base b=4b = 4b=4 units and area A=16A = 16A=16 square units.

Problem 36

Find the base of a triangle with height 12 cm12 \mathrm{~cm}12 cm more than base and area 32 cm232 \mathrm{~cm}^{2}32 cm2. Options: A. 4 cm4 \mathrm{~cm}4 cm B. 8 cm8 \mathrm{~cm}8 cm C. 12 cm12 \mathrm{~cm}12 cm D. 16 cm16 \mathrm{~cm}16 cm

Problem 37

Three charges of +6.65μC+6.65 \mu \mathrm{C}+6.65μC at the triangle's vertices create a force. Find the resultant force: a. 172.34 N172.34 \mathrm{~N}172.34 N b. 22.98 N22.98 \mathrm{~N}22.98 N c. 34.47 N34.47 \mathrm{~N}34.47 N d. 68.94 N68.94 \mathrm{~N}68.94 N e. 0.89 N0.89 \mathrm{~N}0.89 N

Problem 38

Find angle BBB in triangle ABCABCABC if A=15∘A=15^{\circ}A=15∘ and C=50∘C=50^{\circ}C=50∘.

Problem 39

Find the perimeter of triangle PQR with vertices at P(−10,−2)P(-10,-2)P(−10,−2), Q(−10,6)Q(-10,6)Q(−10,6), and R(5,2)R(5,2)R(5,2).

Problem 40

In triangle $D E F$ , find the longest side given $m \angle D=59^{\circ}, m \angle E=76^{\circ}, m \angle F=45^{\circ}$ .

Find the side lengths of $\triangle ABC$ with ratio $2:3:4$ and perimeter 108.

Find the roller reaction at point A of a symmetric fink roof truss with $s = 3m$ , $h = 1.5m$ , and a vertical load $P = 25kN$ .

Find the sum of the perimeters of the four triangles formed by the diagonals of square $A B C D$ . Choices: A. $2+2 \sqrt{2}$ B. $8+4 \sqrt{2}$ C. $8+8 \sqrt{2}$ D. 16

Find the height $h$ of triangle $ABC$ with area $27 \mathrm{~cm}^2$ and base $AB=6 \mathrm{~cm}$ .

Find the largest perimeter of an isosceles triangle with sides $8.2 \mathrm{~cm}$ and $9.4 \mathrm{~cm}$ measured to the nearest mm.

Determine which sets can be side lengths of a triangle: A. $9,5,15$ B. $10,3,7$ C. $8,5,7$

Find side $BC$ of triangle $ABC$ with $AB=8$ , $AC=8\sqrt{2}$ , $\angle ABC=45^{\circ}$ , $\angle ACB=30^{\circ}$ .

In triangle $ABC$ , with $BC=3\sqrt{2}$ and $\angle ABC=45^{\circ}$ , find the minimum of $CM + MN$ as $M$ and $N$ move on $BD$ and $BC$ .

Identify valid triangle side lengths from these options: A. $16,8,10$ B. $4,12,6$ C. $6,9,17$

If the sides of an equilateral triangle are reduced by $2 \mathrm{~cm}$ , the new perimeter is $\frac{4}{5}$ of the original. Find the original side length.

Sketch two isosceles triangles with one angle $40^{\circ}$ , marking all vertices and angles. Remember, angles sum to $180^{\circ}$ .

Divide a square $ABCD$ with side length $4 \mathrm{~cm}$ into 4 equal triangles using 2 line segments. Explain how.

Find the value of $x$ in a triangle with angles $x, 43^{\circ}, 79^{\circ}$ .

Demonstrați că în triunghiul dreptunghic, lungimea medianei pe ipotenuză este $A O=\frac{B C}{2}$ .

Find the perimeter of triangle ABC where $AF=6$ , $CD=5$ , and $BE=4$ .

Un voilier a un mât de $3,6 \mathrm{~m}$ avec un étai de $4,2 \mathrm{~m}$ et des haubans de $3,9 \mathrm{~m}$ . Est-ce que les triangles VEN et VNT sont rectangles ? Justifie.

Find the length of the third side of a triangle with sides 30.3 m, 18.5 m, and an angle of $61^{\circ}$ between them.

Find the third side of a triangle with sides 30.3 m and 18.5 m, and an angle of $61^{\circ}$ . Options: 24.1 m, 26.8 m, 30.0 m, 32.6 m.

Find the length of the third side of a triangle with sides $600 \mathrm{~m}$ , $250 \mathrm{~m}$ , and opposite angle $34^{\circ}$ .

Find the length of the third side of a triangle with sides $45.3 \mathrm{~m}$ , $29.0 \mathrm{~m}$ , and angle $26^{\circ}$ .

In a coordinate plane, $\triangle ABC$ is a right triangle with $A(-4,1)$ and $B(2,1)$ . Area is 9 sq. units. Find point $C$ .

In a coordinate plane, $\triangle ABC$ has $A(-4,1)$ and $B(2,1)$ with area 9 sq. units. Find other $C$ coordinates besides $(2,4)$ .

Triangle $ABC$ has an area of $108 \, \text{cm}^2$ . If base $x$ and height $y$ are integers, find possible values for $x$ . Choices: (A) 4, (B) 5, (C) 6, (D) 8, (E) 9.

In a triangle, if side $x = \frac{3}{5} z$ and $y = \frac{4}{3} x$ , find the ratio of $y$ to $z$ .

A right triangle has an area of 16 square miles. One leg is 4 miles longer than the other. Find the leg lengths. $b(b+4)=32$ $\frac{1}{2} \times b \times h=16$

Find the height of a scalene triangle with base $b = 4$ units and area $A = 16$ square units.

Find the base of a triangle with height $12 \mathrm{~cm}$ more than base and area $32 \mathrm{~cm}^{2}$ . Options: A. $4 \mathrm{~cm}$ B. $8 \mathrm{~cm}$ C. $12 \mathrm{~cm}$ D. $16 \mathrm{~cm}$

Three charges of $+6.65 \mu \mathrm{C}$ at the triangle's vertices create a force. Find the resultant force: a. $172.34 \mathrm{~N}$ b. $22.98 \mathrm{~N}$ c. $34.47 \mathrm{~N}$ d. $68.94 \mathrm{~N}$ e. $0.89 \mathrm{~N}$

Find angle $B$ in triangle $ABC$ if $A=15^{\circ}$ and $C=50^{\circ}$ .

Find the perimeter of triangle PQR with vertices at $P(-10,-2)$ , $Q(-10,6)$ , and $R(5,2)$ .

Find the perimeter of triangle PQR with vertices at $P(-10,-2)$ , $Q(-10,6)$ , and $R(5,2)$ .

Find the area of the triangle with vertices $(4,4)$ , $(4,0)$ , and $(0,0)$ by counting whole and half squares. 8 sq. units. Which quadrant is the point $(-3,2)$ in? Second.

Prove that $c^{2}=a^{2}+b^{2}-2ab\cos C$ .

Triunghiul echilateral $ABC$ are perimetrul $27 \mathrm{~cm}$ . Găsește lungimea segmentului $MC$ , unde $AN=6 \mathrm{~cm}$ .

Найдите периметр равнобедренного треугольника, если одна сторона $a = 7$ дм, а другая $b = a + 8$ дм.

Una gaviota observa un bote a $42^{\circ}$ de depresión desde 45 m de altura. ¿Cuál es la distancia horizontal al barco?

Desde una torre de 28 m, un salvavidas ve a una persona con un ángulo de depresión de $58^{\circ}$ . ¿Cuál es la distancia a la persona?

已知圆O的半径为2.5，弦AB的长度为2，若三角形ABC为等腰三角形，则求 $BC^{2}=$ 。

Calculați aria triunghiului $B O D$ și măsura unghiului $B A D$ în patrulaterul convex $A B C D$ cu datele date.

4. Într-un pătrat $A B C D$ , cu punctele $E, F, G$ pe laturi, arată că triunghiul $D E G$ este isoscel și află unghiul dintre $D F$ și $E G$ .

5. Într-un dreptunghi cu $AB=12 \mathrm{~cm}$ și $BC=4 \mathrm{~cm}$ , calculează aria triunghiului $CEO$ și unghiul $BFC$ .

Un topógrafo mide el ángulo de elevación al cerro: $45^{\circ}$ y a $800$ m más, $30^{\circ}$ . ¿Cuál es la altura del cerro?

Calculate the distance $BC$ in triangle $ABC$ where $AB=1.5 \text{ km}$ , $AC=1.2 \text{ km}$ , and $\angle A=50^\circ$ .

Find the perimeter of triangle $ABC$ with $AD=4 \sqrt{3} cm$ , $BC=5 cm$ , and $BD=1 cm$ .

Find the expression for the area of a triangular attic floor with height $n \mathrm{yd}$ and base $6 \mathrm{yd}$ .

삼각형의 변이 $x-1, x, x+1$ 일 때, $x$ 의 가능한 값 중 아닌 것은? (1) 2 (2) 3 (3) 4 (4) 5 (5) 6

Calcola perimetro e area di un triangolo rettangolo con cateti di $10 \mathrm{~cm}$ e $24 \mathrm{~cm}$ .

Calcola il perimetro e l'area di un triangolo rettangolo con ipotenusa $75 \mathrm{~cm}$ e cateto $45 \mathrm{~cm}$ .

Calcola perimetro e area di un triangolo rettangolo con ipotenusa $75 \mathrm{~cm}$ e cateto $45 \mathrm{~cm}$ .

Un triangolo rettangolo ha area $180 \mathrm{~cm}^{2}$ e un cateto di $9 \mathrm{~cm}$ . Trova il perimetro.

Berechne die fehlende Kathetenlänge $b$ eines rechtwinkligen Dreiecks mit Fläche $A=490mm^2$ und Kathete $a=28mm$ .

Gegeben sind $A=5 \, \text{cm}^{2}$ , $a=80 \, \text{mm}$ und $\gamma=90^{\circ}$ . Welche geometrische Figur ist gefragt?

Berechne die fehlende Kathetenlänge eines rechtwinkligen Dreiecks mit $\gamma=90^{\circ}$ , Fläche 5 cm $^2$ und Kathete 80 mm.

Bài 1. Cho tam giác vuông $\mathrm{ABC}$ với $\mathrm{AB}=30 \mathrm{~cm}$ , cao $\mathrm{AH}=24 \mathrm{cm}$ . Tính $\mathrm{BH}, \mathrm{BC}, \mathrm{AC}$ và độ dài $\mathrm{HD}$ với $\mathrm{HD} \perp \mathrm{AC}$ .

Un triangolo isoscele ha base $32 \mathrm{~cm}$ e altezza $12 \mathrm{~cm}$ . Trova perimetro e area.

Calcola l'area di un triangolo isoscele con perimetro $36 \mathrm{~cm}$ e lato obliquo $13 \mathrm{~cm}$ .

Diberi $AB = AC$ dan kecerunan $AB = \frac{2}{3}$ . Jika $BC = 12$ cm, cari tinggi $A$ dari $BC$ .

Find the area of a triangle with a chord of length $10 \mathrm{~cm}$ in a circle of radius $13 \mathrm{~cm}$ .

In un triangolo rettangolo, i cateti sommano a $21 \mathrm{~cm}$ e differiscono di $3 \mathrm{~cm}$ . Trova l'altezza sull'ipotenusa di $15 \mathrm{~cm}$ .

In un triangolo rettangolo, i cateti sommano a $21 \mathrm{~cm}$ e differiscono di $3 \mathrm{~cm}$ . Trova l'altezza sull'ipotenusa di $15 \mathrm{~cm}$ .

Calcola il perimetro di un triangolo isoscele con area di $672 \mathrm{~cm}^{2}$ e base di $28 \mathrm{~cm}$ .

Un triangolo ha area $14,95 \mathrm{~cm}^{2}$ e un cateto di $6,5 \mathrm{~cm}$ . Trova l'altro cateto e l'altezza sull'ipotenusa di $13 \mathrm{~cm}$ .

In un triangolo rettangolo, un cateto è $36 \mathrm{~cm}$ e l'altro è $\frac{4}{3}$ volte il primo. Trova perimetro e area.