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Congruent
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Triangle
Trigonometry
Surface area and Volume
Probability
100

Are two squares with the same size congruent?


Yes, they are congruent because they have the same shape and size.

100

Triangle A has sides 6 cm, 8 cm, and 10 cm.

Triangle B has its shortest side equal to 9 cm.

What are the lengths of the other two sides of Triangle B?

The other sides are 12cm and 15 cm

100

In triangle ABC, angle A = 30°, and the side opposite angle A (BC) = 10 cm. Find the hypotenuse (AC) assuming it’s a right triangle at B.


Area formula:

\text{Area} = \frac{1}{2}ab\sin(C)


= \frac{1}{2}(8)(6)\sin(45°) = 24 \times 0.7071 \approx 16.97 \text{ cm}^2


100

Find the volume of a sphere with radius 7 cm

Calculation:


=(4/3) × 3.14×73

100

A point is chosen at random inside a square of side length 4 units. What is the probability that it lands inside a circle of radius 2 units inscribed in the square?



  • Area of square: 4 \times 4 = 16
  • Area of circle: \pi \times 2^2 = 4\pi
  • Probability:
    P = \frac{4\pi}{16} = \frac{\pi}{4}


200

Rectangle A has a length of 6 cm and a width of 3 cm.

Rectangle B has a length of 6 cm and a width of 3 cm.

Are the two rectangles congruent?

Yes, the rectangles are congruent because they have the same shape and exact same side lengths.

200

A map uses a scale where 1 inch

Two towns are 3 inches apart on the map.

How many miles apart are the towns in real life?

The towns are 15miles apart

200

In a right triangle, one of the acute angles is 35°, and the hypotenuse is 12 cm. Find the length of the side opposite the 35° angle.



Answer: The opposite side is approximately 6.89 cm long.


200

A cone has a radius of 3 cm and a height of 9 cm

Find its volume

(Use TT = 3.14)

=(1/3) x3.14x32x9

200

A point is randomly selected inside a triangle with area 20 square units. Inside the triangle is a smaller triangle with area 5 square units.

What is the probability that the point lands inside the smaller triangle?


Solution:


P = \frac{\text{Area of smaller triangle}}{\text{Area of larger triangle}} = \frac{5}{20} = \boxed{\frac{1}{4}}


300

Are these two triangles congruent?


Triangle A has sides 3 cm, 4 cm, and 5 cm.

Triangle B has sides 3 cm, 4 cm, and 5 cm.


Yes, the two triangles are congruent because all three sides are the same length.

300

A side of the smaller polygon is 8 cm.

What is the length of the corresponding side in the larger polygon?

The corresponding side is 20 cm

300

In triangle ABC, angle A = 40°, angle B = 60°, and side a = 10 cm. Find side b.


Answer:

b ≈ 13.47 cm


300

Find the surface area of a cube with side

=6x62

6 cm

=6x36

= 216 cm?

300

A square has a side length of 6 units. A circle of radius 3 units is drawn inside the square so that it touches all four sides (it’s inscribed).

What is the probability that a randomly chosen point inside the square also lies inside the circle



  • Area of square: 6 \times 6 = 36
  • Area of circle: \pi \times 3^2 = 9\pi
  • Probability:
    P = \frac{9\pi}{36} = \frac{\pi}{4}


400

Quadrilateral ABCD is congruent to quadrilateral WXYZ. If side AB = 7 cm, angle B = 90°, and side CD = 5 cm, what are the measures of side WX, angle X, and side YZ?


  • Side WX = Side AB = 7 cm
  • Angle X = Angle B = 90°
  • Side YZ = Side CD = 5 cm


400

The dimensions of Rectangle A are 4 cm by 6

CT.

The length of Rectangle B is 10 cm.

What is the width of Rectangle B?

The width of rectangle B is 6.67 cm

400

In triangle XYZ, side x = 7 cm, y = 9 cm, z = 10 cm. Find angle X.


Answer:

X ≈ 42.8°


400

Find the surface area of a rectangular prism with dimensions:

length = 8 cm, width = 5 cm, height = 3

cm

Calculation:

=2(8x5+8x3+5x3)

=2(40 + 24 + 15)

= 2(79)

= 158 cm2

400

A circle with radius 5 units is drawn. A square is inscribed in the circle (all four vertices touch the circle).

What is the probability that a randomly chosen point inside the circle also lies inside the square?



  • Radius of circle = 5
  • Area of circle = \pi \times 5^2 = 25\pi
  • Side length of inscribed square = 5\sqrt{2} \times 2 = 10\sqrt{2}, but that’s too long. Let’s fix it properly:


500

Two triangles, ΔABC and ΔDEF, are congruent. If angle A = 40°, angle B = 60°, and side AB = 5 cm, what are the measures of angles D and E, and the length of side DE?


  • Angle D = Angle A = 40°
  • Angle E = Angle B = 60°
  • Side DE = Side AB = 5 cm


500

The area of the smaller triangle is 27 cm?.

What is the area of the larger triangle?

(Hint: Areas scale by the square of the scale factor)

The area of the larger triangle is 75cm2

500

In a right triangle, angle A = 30°, hypotenuse = 10 cm. Find the length of the side opposite angle A.


Answer:

Opposite side = 5 cm


500

Find the volume of a cylinder with radius = 4

cm and height = 10 cm

(Use nt = 3.14)

Calculation:

=3.14 × 42 × 10

= 3.14 × 16 × 10

= 3.14 × 160

= 502.4 cmg

500

A point is chosen at random inside a rectangle that is 10 units long and 4 units wide. A circle with radius 2 units is drawn completely inside the rectangle.

What is the probability that the point lands inside the circle?



  • Area of rectangle = 10 \times 4 = 40
  • Area of circle = \pi \times 2^2 = 4\pi
  • Probability = \frac{4\pi}{40} = \frac{\pi}{10}