Diagram & Picture

Problem 1

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

See Solution

Problem 2

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 3

Prove that in a given diagram, (i) $A B$ is parallel to $P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 4

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 5

Prove that if $B P$ and $P Q$ are tangents to circles, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

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Problem 6

If $\overline{F B} \| \overline{E C}$ and $m \angle E F B=103^{\circ}$ , find $m \angle C E F$ .

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Problem 7

A mud house has a cone roof over a cylinder. Given dimensions, find $A B$ , volume, surface area, and house choice.

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Problem 8

Draw a Venn diagram for 400 Grade 7 students: 200 play ML only, 85 play COD only, and 50 play neither.

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Problem 9

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 10

Tentukan nilai nombor $4025_{6}$ dalam basis sepuluh.

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Problem 11

Find the area of a square (side 2) with a semi-circle on top. Options: a) $6+\pi$ , b) $2+\pi$ , c) $4+4\pi$ , d) $4+\frac{\pi}{2}$ , e) $4+\pi$ .

See Solution

Problem 12

Draw a Venn diagram for 400 Grade 7 students: 200 play ML only, 85 play COD only, and 50 play neither.

See Solution

Problem 13

In parallelogram $A B C D$ , with $A B = B E = E C$ and area of triangle $B E C$ as 8, find the perimeter of polygon $A B E C D$ .

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Problem 14

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 15

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 16

Find the area of parallelogram $ABCD$ with sides 6 and 4 units and an angle of 125 degrees. Compare it to 24.

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Problem 17

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

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Problem 18

Desenați: a) dreapta $d$ , punctul $A$ exterior, și punctele $C$ , $D$ pe $d$ ; b) dreapta $d$ , punctul $M$ pe $d$ și colorați 2 semidrepte; c) 3 semidrepte cu originea $O$ ; d) un plan cu 2 semiplane, colorând unul.

See Solution

Problem 19

Kotak mengandungi butang putih, hitam, kuning, dan biru yang sama bilangan. Jika totalnya 123, kira jumlah butang putih, hitam, dan kuning.

See Solution

Problem 20

Tukarkan $2 \frac{7}{10}$ kepada pecahan tak waiar.

See Solution

Problem 21

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.

See Solution

Problem 22

Vẽ tia $\mathrm{Ax}$ , chọn $\mathrm{B} \in \mathrm{Ax}$ với $\mathrm{AB}=8 \mathrm{~cm}$ , và $\mathrm{C}$ sao cho $\mathrm{AC}=4 \mathrm{~cm}$ .
a) C có nằm giữa A và B không? Tại sao? b) So sánh $\mathrm{CA}$ và $\mathrm{CB}$ . c) C có phải là trung điểm của $\mathrm{AB}$ không? Tại sao?

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Problem 23

If $\overline{D E} \| \overline{A C}$ , which proportion is true? A) $\frac{D A}{D E}=\frac{E C}{D E}$ B) $\frac{A C}{D E}=\frac{E C}{B E}$ C) $\frac{B E}{D E}=\frac{E C}{A C}$ D) $\frac{B E}{B C}=\frac{D E}{A C}$

See Solution

Problem 24

Find the length of TA where $OT = 4 \mathrm{~cm}$ and $\angle OTA = 30^{\circ}$ . Choices: (a) $2\sqrt{3} \mathrm{~cm}$ (b) $2 \mathrm{~cm}$ (c) $2\sqrt{2} \mathrm{~cm}$ (d) $\sqrt{3} \mathrm{~cm}$ .

See Solution

Problem 25

Find the refractive index $\mu$ of a sphere with radius $R$ and a cavity of radius $R/2$ focusing light at point P. Options: (a) $\mu=\frac{3+\sqrt{5}}{2}$ , (b) $\mu=\frac{3-\sqrt{5}}{2}$ , (c) $\mu=3+\sqrt{5}$ , (d) $\mu=\frac{1+\sqrt{5}}{2}$ .

See Solution

Problem 26

Graph the points -2, 4, and 14 on the number line.

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Problem 27

Toss a fair coin and a four-sided die. Find the probability of heads and an even die result. Complete the diagram.

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Problem 28

Find the coordinates of point $S$ in parallelogram $P Q R S$ with $P(-1,3)$ , $Q(-2,-3)$ , $R(1,0)$ . Also, find the equation of the perpendicular bisector of $P R$ and determine if $P Q R S$ is a rhombus.

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Problem 29

Find the equation of line $O A$ , show $A$ is $(16,-8)$ , and calculate area of trapezium $O A B C$ .

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Problem 30

Find the equation of line $O A$ , show $A$ is at $(16,-8)$ , and calculate the area of trapezium $O A B C$ .

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Problem 31

A 3 m composite beam with a 30 kN/m load has wood (100 mm x 150 mm) and an 8 mm steel plate. Find max bending stresses $\sigma_{\infty}$ and $\sigma$ for wood and steel with $E_{z}=10$ GPa and $E_{s}=210$ GPa.

See Solution

Problem 32

Find the area of a shaded sector in a circle with radius 16 units and central angle $60^{\circ}$ .

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Problem 33

Locate points of rectangle $ABCD$ at $A(2,2), B(2,-1), C(-1,-1), D(-1,2)$ , then find its perimeter and area.

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Problem 34

在 $\square A B C D$ 中, $A C, B D$ 交于点 $O$ , 点 $E, F$ 在 $A C$ 上, $A E=C F$ 。证明四边形 $E B F D$ 是平行四边形和菱形。

See Solution

Problem 35

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

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Problem 36

Cari nilai $x$ jika $n(\xi)=52, n(P)=40, n(P \cap Q)=9$ dan $n\left(P^{\prime}\right)=x+7$ .

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Problem 37

Постройте угол bisectrix с помощью циркуля и линейки.

See Solution

Problem 38

Find $m \angle Q O R$ if $m \angle P O Q=24$ and $m \angle P O R=59$ .

See Solution

Problem 39

In square $ABCD$ , area equals the sum of areas of triangles $ABE$ and $DCE$ . If $AB=6$ , find $CE$ . (a) 5 (b) 6 (c) 2 (d) 3 (e) 4

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Problem 40

1. Într-un triunghi dreptunghic $A B C$ cu $\angle C=60^{\circ}$ , găsiți măsura unghiului $A D C$ (bisectoarea $B A M$ ).
2. Calculați aria unui paralelogram cu laturile $A B=170 \mathrm{~m}, B C=80 \mathrm{~m}$ și diagonala $B D=150 \mathrm{~m}$ .

See Solution

Problem 41

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 42

Place the point on the number line to represent the value of $\frac{8}{6}$ .

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Problem 43

Complete the area model and then calculate $24 \times 35$ using the standard algorithm.

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Problem 44

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 45

Identify the two whole numbers between which $\sqrt{54}$ lies and justify your answer.

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Problem 46

Determine the two whole numbers between which $\sqrt{98}$ lies. Justify your answer and position $\sqrt{98}$ on the number line.

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Problem 47

Find the two whole numbers between which $\sqrt{77}$ lies. Justify your answer and place $\sqrt{77}$ on a number line.

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Problem 48

Determine the two consecutive whole numbers that $\sqrt{10}$ falls between. Explain your reasoning.

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Problem 49

Find the two consecutive whole numbers between which $\sqrt{93}$ lies.

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Problem 50

Find two consecutive whole numbers that $\sqrt{52}$ is between and justify your answer.

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Problem 51

Find the two whole numbers between which $\sqrt{87}$ lies. Justify your answer and place $\sqrt{87}$ on a number line.

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Problem 52

Find the two whole numbers between which $\sqrt{12}$ lies and justify your answer.

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Problem 53

Find two consecutive whole numbers between which $\sqrt{85}$ lies. Justify your answer and locate it on a number line.

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Problem 54

Find two whole numbers between which $\sqrt{92}$ falls. Justify your answer and place $\sqrt{92}$ on a number line.

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Problem 55

Determine the two whole numbers that $\sqrt{50}$ is between and justify your answer.

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Problem 56

Find the two whole numbers between which $\sqrt{40}$ lies. Fill in the blanks: $\sqrt{40}$ is between _ and _.

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Problem 57

Determine the two whole numbers between which $\sqrt{38}$ falls.

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Problem 58

Determine the two consecutive whole numbers that $\sqrt{86}$ falls between and justify your answer.

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Problem 59

Does $4 \div \frac{3}{4}$ equal $5 \frac{1}{4}$ ? Discuss Lin's claim about groups of $\frac{3}{4}$ and leftovers.

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Problem 60

Sebuah papan sasar bulatan dengan jejari $20 \mathrm{~cm}$ . Beza jejari antara bulatan $P, Q, R, S$ dan $T$ ialah $4 \mathrm{~cm}$ .
Hitung: (a) Beza lilitan bulatan $T$ dan $P$ . (b) Luas kawasan $P$ dan $R$ (gunakan $\pi=\frac{22}{7}$ ).

See Solution

Problem 61

Hitung beza lilitan bulatan $T$ dan $P$ , serta luas kawasan $P$ dan $R$ dengan $\pi=\frac{22}{7}$ .

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Problem 62

Rappresenta con un diagramma di Eulero-Venn gli insiemi: a. $A=\{x \mid x$ è un giorno della settimana $\}$ b. $B=\{x \mid x$ è un mese che inizia con $A\}$ c. $C=\{x \mid x$ è una cifra di 96.820 $\}$

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Problem 63

Given sets $X=\{1,2,3\}$ and $Y=\{13,31,65,23\}$ , do the following:
1. Write the relation $R$ and draw an arrow diagram.
2. Determine if $2 R 65$ , $1 R 31$ , $3 R 13$ are true, with reasons.
3. List $M=\{(y, 23):(y, 23) \in R\}$ .

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Problem 64

How do you use a number line to show the subtraction of two integers, say $a - b$ ?

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Problem 65

In a survey of 100 traders selling fruits, find those selling only oranges and mangoes using a Venn diagram.

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Problem 66

Draw $\mathrm{H}_{2} \mathrm{O}$ and $\mathrm{CH}_{4}$ , showing valence electrons, bond types, and partial charges. Describe their reaction.

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Problem 67

Find angle $O Q P$ given points $P, Q, R$ on a circle with $m\angle Q P R=35^{\circ}$ and $m\angle O R P=30^{\circ}$ .

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Problem 68

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 69

Solve for $x$ in the equation $2 \sin x = \sqrt{3} \tan x$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

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Problem 70

Find the equation of line $PQ$ , check if $PS$ is perpendicular to $PQ$ , find $QR$ parallel to $PS$ , calculate angle $\theta$ , and distance $QS$ .

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Problem 71

Find the angle $\theta$ at the center of sector $OAB$ with radius $9 \mathrm{~cm}$ and arc length \overparen{AB}=6 \pi \mathrm{~cm}.

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Problem 72

In sectors $O A B$ and $O C D$ , with $O A=27 \mathrm{~cm}$ , $O C=15 \mathrm{~cm}$ , and shaded area $147 \pi \mathrm{cm}^{2}$ , find: (a) $\angle A O B$ ; (b) perimeter of shaded region in terms of $\pi$ .

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Problem 73

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 74

Find the equation of line $PQ$ for quadrilateral $PQRS$ with points $P(-2,1)$ , $Q(1,-5)$ , and $S(2,3)$ .

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Problem 75

Locate 256 on the number line. Is it closer to 250 or 260? Round 256 to the nearest ten.

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Problem 76

Show three ways to represent the number 4.672 using a place-value chart with digits.

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Problem 77

Find the prime factors of 72 and 18 using the ladder method and multiply the factors on the left side.

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Problem 78

Find the line of reflection for triangle $XYZ$ with $Z'(10,8)$ given $X(4,-5)$ , $Y(6,-1)$ , $Z(10,-8)$ .

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Problem 79

Show $\frac{5}{3} \times \frac{3}{4}$ on a number line. Is it less than, greater than, or equal to $\frac{3}{4}$ ?

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Problem 80

Fill in the chart showing powers of ten division with 9 in 'Ones' and 0 in 'Tenths', 'Hundredths', 'Thousandths'.
Ones: 9, Tenths: 0, Hundredths: 0, Thousandths: 0.

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Problem 81

Model $4 + (-9)$ on the number line.

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Problem 82

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 83

Compare the value of the 6 in the thousands place ( $6000$ ) to the 6 in the hundreds place ( $600$ ).

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Problem 84

Find the values of $a$ , $b$ , $c$ , and $d$ in the magic square where rows, columns, and diagonals sum to the same number.

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Problem 85

Complete the unmagic square with different sums for rows, columns, and diagonals using digits 1-9:
8 & & 9 & 7 & 1 & & 6
Choose from options A, B, C, or D.

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Problem 86

Find the height corresponding to side $S P$ of parallelogram $P Q R S$ if reducing sides $S P$ and $R Q$ by $10 \mathrm{~cm}$ decreases area by $120 \mathrm{~cm}^{2}$ .

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Problem 87

Find the value of $z$ given $x = z$ , $x = 6k + 13$ , and $y = 8k - 29$ with lines $m$ and $n$ parallel.

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Problem 88

Create subitizing cards for these expressions and find the solutions:
1. $-3+7$
2. $-4+4$
3. $5+(-7)$

See Solution

Problem 89

Draw the angle $150^{\circ}$ , find its reference angle, and identify the quadrant of its terminal side.

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Problem 90

Draw the angle $-240^{\circ}$ in standard position and identify its quadrant.

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Problem 91

Subtract $9 \frac{1}{2} - 8 \frac{5}{8}$ and simplify the answer to a whole number, proper fraction, or mixed number.

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Problem 92

Complete the equation using the place value chart: 10 ÷ 10 = $$.

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Problem 93

Divide 192 by a 2-digit number 16 and illustrate with a simple drawing. Find the result of $\frac{192}{16}$ .

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Problem 94

10. Identify pairs of vertically opposite, adjacent, linear pair, complementary, and supplementary angles. Given $\angle 4=110^{\circ}$ and $\angle 5=120^{\circ}$ , find the others.
11. What is the angle that equals its complement?
12. What is the angle that equals its supplement?

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Problem 95

Draw 2 line segments of different lengths with $M$ as their midpoint.

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Problem 96

Given $A D \parallel B C$ , with $A D=35 \mathrm{~cm}$ , $B D=37 \mathrm{~cm}$ , and $B C=45 \mathrm{~cm}$ , find $A B$ and the area of trapezium $A B C D$ .

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Problem 97

Given $AB=36 \mathrm{~cm}, BC=48 \mathrm{~cm}, AD=109 \mathrm{~cm}$ , find $AC$ , $CD$ , and the area of quadrilateral $ABCD$ .

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Problem 98

In line $B D C$ , given $A B=40$ , $B C=32$ , $A C=24$ , and $C D=7$ , determine if $\angle C$ is right and find $A D$ .

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Problem 99

Find the area of triangle $ABC$ given $AE=10 \mathrm{~cm}, BC=20 \mathrm{~cm}, CD=17 \mathrm{~cm}, DE=2 \mathrm{~cm}, EF=6 \mathrm{~cm}$ , correct to 3 significant figures.

See Solution

Problem 100

Which diagram shows "If it is a triangle, then it has three vertices"? A. 3 vertices outside B. 3 vertices inside

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Diagram & Picture

Problem 1

Given a circle with center OOO and diameter MNMNMN, prove:(i) ∠MAN=90∘+∠PQR\angle MAN = 90^{\circ} + \angle PQR∠MAN=90∘+∠PQR,(ii) ∠QPR+2×∠MAN=360∘\angle QPR + 2 \times \angle MAN = 360^{\circ}∠QPR+2×∠MAN=360∘.

Problem 2

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 3

Prove that in a given diagram, (i) AB A B AB is parallel to PQ P Q PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 4

Prove that if BP B P BP and PQ P Q PQ are tangents, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 5

Prove that if BP B P BP and PQ P Q PQ are tangents to circles, then (i) AB∥PQ A B \parallel P Q AB∥PQ and (ii) MP×AM=BM×MQ M P \times A M = B M \times M Q MP×AM=BM×MQ.

Problem 6

If FB‾∥EC‾ \overline{F B} \| \overline{E C} FB∥EC and m∠EFB=103∘ m \angle E F B=103^{\circ} m∠EFB=103∘, find m∠CEF m \angle C E F m∠CEF.

Problem 7

A mud house has a cone roof over a cylinder. Given dimensions, find AB A B AB, volume, surface area, and house choice.

Problem 8

Draw a Venn diagram for 400 Grade 7 students: 200 play ML only, 85 play COD only, and 50 play neither.

Problem 9

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 10

Tentukan nilai nombor 402564025_{6}40256​ dalam basis sepuluh.

Problem 11

Find the area of a square (side 2) with a semi-circle on top. Options: a) 6+π6+\pi6+π, b) 2+π2+\pi2+π, c) 4+4π4+4\pi4+4π, d) 4+π24+\frac{\pi}{2}4+2π​, e) 4+π4+\pi4+π.

Problem 12

Draw a Venn diagram for 400 Grade 7 students: 200 play ML only, 85 play COD only, and 50 play neither.

Problem 13

In parallelogram ABCDA B C DABCD, with AB=BE=ECA B = B E = E CAB=BE=EC and area of triangle BECB E CBEC as 8, find the perimeter of polygon ABECDA B E C DABECD.

Problem 14

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 15

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

Problem 16

Find the area of parallelogram ABCDABCDABCD with sides 6 and 4 units and an angle of 125 degrees. Compare it to 24.

Problem 17

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

Problem 18

Desenați: a) dreapta ddd, punctul AAA exterior, și punctele CCC, DDD pe ddd; b) dreapta ddd, punctul MMM pe ddd și colorați 2 semidrepte; c) 3 semidrepte cu originea OOO; d) un plan cu 2 semiplane, colorând unul.

Problem 19

Kotak mengandungi butang putih, hitam, kuning, dan biru yang sama bilangan. Jika totalnya 123, kira jumlah butang putih, hitam, dan kuning.

Problem 20

Tukarkan 27102 \frac{7}{10}2107​ kepada pecahan tak waiar.

Problem 21

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 22

Problem 23

Problem 24

Find the length of TA where OT=4 cmOT = 4 \mathrm{~cm}OT=4 cm and ∠OTA=30∘\angle OTA = 30^{\circ}∠OTA=30∘. Choices: (a) 23 cm2\sqrt{3} \mathrm{~cm}23​ cm (b) 2 cm2 \mathrm{~cm}2 cm (c) 22 cm2\sqrt{2} \mathrm{~cm}22​ cm (d) 3 cm\sqrt{3} \mathrm{~cm}3​ cm.

Problem 25

Problem 26

Graph the points -2, 4, and 14 on the number line.

Problem 27

Toss a fair coin and a four-sided die. Find the probability of heads and an even die result. Complete the diagram.

Problem 28

Find the coordinates of point SSS in parallelogram PQRSP Q R SPQRS with P(−1,3)P(-1,3)P(−1,3), Q(−2,−3)Q(-2,-3)Q(−2,−3), R(1,0)R(1,0)R(1,0). Also, find the equation of the perpendicular bisector of PRP RPR and determine if PQRSP Q R SPQRS is a rhombus.

Problem 29

Find the equation of line OAO AOA, show AAA is (16,−8)(16,-8)(16,−8), and calculate area of trapezium OABCO A B COABC.

Problem 30

Find the equation of line OAO AOA, show AAA is at (16,−8)(16,-8)(16,−8), and calculate the area of trapezium OABCO A B COABC.

Problem 31

A 3 m composite beam with a 30 kN/m load has wood (100 mm x 150 mm) and an 8 mm steel plate. Find max bending stresses σ∞\sigma_{\infty}σ∞​ and σ\sigmaσ for wood and steel with Ez=10E_{z}=10Ez​=10 GPa and Es=210E_{s}=210Es​=210 GPa.

Problem 32

Find the area of a shaded sector in a circle with radius 16 units and central angle 60∘60^{\circ}60∘.

Problem 33

Locate points of rectangle ABCDABCDABCD at A(2,2),B(2,−1),C(−1,−1),D(−1,2)A(2,2), B(2,-1), C(-1,-1), D(-1,2)A(2,2),B(2,−1),C(−1,−1),D(−1,2), then find its perimeter and area.

Problem 34

在 □ABCD\square A B C D□ABCD 中, AC,BDA C, B DAC,BD 交于点 OOO, 点 E,FE, FE,F 在 ACA CAC 上, AE=CFA E=C FAE=CF。证明四边形 EBFDE B F DEBFD 是平行四边形和菱形。

Problem 35

Triunghiul echilateral ABCABCABC are perimetrul 27 cm27 \mathrm{~cm}27 cm. Găsește lungimea segmentului MCMCMC, unde AN=6 cmAN=6 \mathrm{~cm}AN=6 cm.

Problem 36

Cari nilai xxx jika n(ξ)=52,n(P)=40,n(P∩Q)=9n(\xi)=52, n(P)=40, n(P \cap Q)=9n(ξ)=52,n(P)=40,n(P∩Q)=9 dan n(P′)=x+7n\left(P^{\prime}\right)=x+7n(P′)=x+7.

Problem 37

Постройте угол bisectrix с помощью циркуля и линейки.

Problem 38

Find m∠QORm \angle Q O Rm∠QOR if m∠POQ=24m \angle P O Q=24m∠POQ=24 and m∠POR=59m \angle P O R=59m∠POR=59.

Problem 39

In square ABCDABCDABCD, area equals the sum of areas of triangles ABEABEABE and DCEDCEDCE. If AB=6AB=6AB=6, find CECECE. (a) 5 (b) 6 (c) 2 (d) 3 (e) 4

Problem 40

Problem 41

4. Într-un pătrat ABCDA B C DABCD, cu punctele E,F,GE, F, GE,F,G pe laturi, arată că triunghiul DEGD E GDEG este isoscel și află unghiul dintre DFD FDF și EGE GEG.

Problem 42

Given a circle with center $O$ and diameter $MN$ , prove:
(i) $\angle MAN = 90^{\circ} + \angle PQR$ ,
(ii) $\angle QPR + 2 \times \angle MAN = 360^{\circ}$ .

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Prove that in a given diagram, (i) $A B$ is parallel to $P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Prove that if $B P$ and $P Q$ are tangents, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

Prove that if $B P$ and $P Q$ are tangents to circles, then (i) $A B \parallel P Q$ and (ii) $M P \times A M = B M \times M Q$ .

If $\overline{F B} \| \overline{E C}$ and $m \angle E F B=103^{\circ}$ , find $m \angle C E F$ .

A mud house has a cone roof over a cylinder. Given dimensions, find $A B$ , volume, surface area, and house choice.

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

Tentukan nilai nombor $4025_{6}$ dalam basis sepuluh.

Find the area of a square (side 2) with a semi-circle on top. Options: a) $6+\pi$ , b) $2+\pi$ , c) $4+4\pi$ , d) $4+\frac{\pi}{2}$ , e) $4+\pi$ .

In parallelogram $A B C D$ , with $A B = B E = E C$ and area of triangle $B E C$ as 8, find the perimeter of polygon $A B E C D$ .

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 area of parallelogram $ABCD$ with sides 6 and 4 units and an angle of 125 degrees. Compare it to 24.

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

Desenați: a) dreapta $d$ , punctul $A$ exterior, și punctele $C$ , $D$ pe $d$ ; b) dreapta $d$ , punctul $M$ pe $d$ și colorați 2 semidrepte; c) 3 semidrepte cu originea $O$ ; d) un plan cu 2 semiplane, colorând unul.

Tukarkan $2 \frac{7}{10}$ kepada pecahan tak waiar.

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 TA where $OT = 4 \mathrm{~cm}$ and $\angle OTA = 30^{\circ}$ . Choices: (a) $2\sqrt{3} \mathrm{~cm}$ (b) $2 \mathrm{~cm}$ (c) $2\sqrt{2} \mathrm{~cm}$ (d) $\sqrt{3} \mathrm{~cm}$ .

Find the coordinates of point $S$ in parallelogram $P Q R S$ with $P(-1,3)$ , $Q(-2,-3)$ , $R(1,0)$ . Also, find the equation of the perpendicular bisector of $P R$ and determine if $P Q R S$ is a rhombus.

Find the equation of line $O A$ , show $A$ is $(16,-8)$ , and calculate area of trapezium $O A B C$ .

Find the equation of line $O A$ , show $A$ is at $(16,-8)$ , and calculate the area of trapezium $O A B C$ .

A 3 m composite beam with a 30 kN/m load has wood (100 mm x 150 mm) and an 8 mm steel plate. Find max bending stresses $\sigma_{\infty}$ and $\sigma$ for wood and steel with $E_{z}=10$ GPa and $E_{s}=210$ GPa.

Find the area of a shaded sector in a circle with radius 16 units and central angle $60^{\circ}$ .

Locate points of rectangle $ABCD$ at $A(2,2), B(2,-1), C(-1,-1), D(-1,2)$ , then find its perimeter and area.

在 $\square A B C D$ 中, $A C, B D$ 交于点 $O$ , 点 $E, F$ 在 $A C$ 上, $A E=C F$ 。证明四边形 $E B F D$ 是平行四边形和菱形。

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

Cari nilai $x$ jika $n(\xi)=52, n(P)=40, n(P \cap Q)=9$ dan $n\left(P^{\prime}\right)=x+7$ .

Find $m \angle Q O R$ if $m \angle P O Q=24$ and $m \angle P O R=59$ .

In square $ABCD$ , area equals the sum of areas of triangles $ABE$ and $DCE$ . If $AB=6$ , find $CE$ . (a) 5 (b) 6 (c) 2 (d) 3 (e) 4

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 $E G$ .

Place the point on the number line to represent the value of $\frac{8}{6}$ .

Complete the area model and then calculate $24 \times 35$ using the standard algorithm.

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

Identify the two whole numbers between which $\sqrt{54}$ lies and justify your answer.

Determine the two whole numbers between which $\sqrt{98}$ lies. Justify your answer and position $\sqrt{98}$ on the number line.

Find the two whole numbers between which $\sqrt{77}$ lies. Justify your answer and place $\sqrt{77}$ on a number line.

Determine the two consecutive whole numbers that $\sqrt{10}$ falls between. Explain your reasoning.

Find the two consecutive whole numbers between which $\sqrt{93}$ lies.

Find two consecutive whole numbers that $\sqrt{52}$ is between and justify your answer.

Find the two whole numbers between which $\sqrt{87}$ lies. Justify your answer and place $\sqrt{87}$ on a number line.

Find the two whole numbers between which $\sqrt{12}$ lies and justify your answer.

Find two consecutive whole numbers between which $\sqrt{85}$ lies. Justify your answer and locate it on a number line.

Find two whole numbers between which $\sqrt{92}$ falls. Justify your answer and place $\sqrt{92}$ on a number line.

Determine the two whole numbers that $\sqrt{50}$ is between and justify your answer.

Find the two whole numbers between which $\sqrt{40}$ lies. Fill in the blanks: $\sqrt{40}$ is between _ and _.

Determine the two whole numbers between which $\sqrt{38}$ falls.

Determine the two consecutive whole numbers that $\sqrt{86}$ falls between and justify your answer.

Does $4 \div \frac{3}{4}$ equal $5 \frac{1}{4}$ ? Discuss Lin's claim about groups of $\frac{3}{4}$ and leftovers.

Sebuah papan sasar bulatan dengan jejari $20 \mathrm{~cm}$ . Beza jejari antara bulatan $P, Q, R, S$ dan $T$ ialah $4 \mathrm{~cm}$ .
Hitung: (a) Beza lilitan bulatan $T$ dan $P$ . (b) Luas kawasan $P$ dan $R$ (gunakan $\pi=\frac{22}{7}$ ).

Hitung beza lilitan bulatan $T$ dan $P$ , serta luas kawasan $P$ dan $R$ dengan $\pi=\frac{22}{7}$ .

Rappresenta con un diagramma di Eulero-Venn gli insiemi: a. $A=\{x \mid x$ è un giorno della settimana $\}$ b. $B=\{x \mid x$ è un mese che inizia con $A\}$ c. $C=\{x \mid x$ è una cifra di 96.820 $\}$

How do you use a number line to show the subtraction of two integers, say $a - b$ ?

Draw $\mathrm{H}_{2} \mathrm{O}$ and $\mathrm{CH}_{4}$ , showing valence electrons, bond types, and partial charges. Describe their reaction.

Find angle $O Q P$ given points $P, Q, R$ on a circle with $m\angle Q P R=35^{\circ}$ and $m\angle O R P=30^{\circ}$ .

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

Solve for $x$ in the equation $2 \sin x = \sqrt{3} \tan x$ for $0^{\circ} \leq x \leq 360^{\circ}$ .

Find the equation of line $PQ$ , check if $PS$ is perpendicular to $PQ$ , find $QR$ parallel to $PS$ , calculate angle $\theta$ , and distance $QS$ .

Find the angle $\theta$ at the center of sector $OAB$ with radius $9 \mathrm{~cm}$ and arc length \overparen{AB}=6 \pi \mathrm{~cm}.

In sectors $O A B$ and $O C D$ , with $O A=27 \mathrm{~cm}$ , $O C=15 \mathrm{~cm}$ , and shaded area $147 \pi \mathrm{cm}^{2}$ , find: (a) $\angle A O B$ ; (b) perimeter of shaded region in terms of $\pi$ .

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

Find the equation of line $PQ$ for quadrilateral $PQRS$ with points $P(-2,1)$ , $Q(1,-5)$ , and $S(2,3)$ .