Geometry Study Resources for Competitive Exams

Geometry

Q. If the radius of a circle is doubled, by what factor does the area increase?

A. 1
B. 2
C. 3
D. 4

Solution

Area = πr^2. If radius is doubled, new area = π(2r)^2 = 4πr^2, so area increases by a factor of 4.

Correct Answer: D — 4

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Q. If the radius of a circle is doubled, by what factor does the area of the circle increase?

A. 1
B. 2
C. 4
D. 8

Solution

The area of a circle is given by A = πr². If the radius is doubled (r' = 2r), the new area A' = π(2r)² = 4πr². Thus, the area increases by a factor of 4.

Correct Answer: C — 4

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Q. If the radius of a circle is doubled, how does the area of the circle change?

A. It remains the same
B. It doubles
C. It triples
D. It quadruples

Solution

The area of a circle is given by A = πr². If the radius is doubled (r' = 2r), the new area A' = π(2r)² = 4πr², which is four times the original area.

Correct Answer: D — It quadruples

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Q. If the radius of a circle is halved, how does the area change?

A. It remains the same
B. It doubles
C. It is halved
D. It is quartered

Solution

If the radius is halved, the area becomes π(r/2)² = π(r²/4) = (1/4) * πr², which means the area is quartered.

Correct Answer: D — It is quartered

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Q. If the radius of a sphere is 3 units, what is its volume?

A. 27π
B. 36π
C. 9π
D. 18π

Solution

The volume V of a sphere is given by V = (4/3)πr³. Thus, V = (4/3)π(3)³ = (4/3)π(27) = 36π.

Correct Answer: A — 27π

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Q. If the radius of a sphere is doubled, how does the volume change?

A. It doubles
B. It triples
C. It quadruples
D. It increases by a factor of eight

Solution

Volume of a sphere = (4/3)πr³. If r is doubled, the new volume = (4/3)π(2r)³ = (4/3)π(8r³) = 8 times the original volume.

Correct Answer: D — It increases by a factor of eight

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Q. If the sides of a triangle are 7 cm, 24 cm, and 25 cm, what is its area?

A. 84 cm²
B. 96 cm²
C. 120 cm²
D. 72 cm²

Solution

Using Heron's formula, s = (7 + 24 + 25)/2 = 28. Area = √(s(s-a)(s-b)(s-c)) = √(28(28-7)(28-24)(28-25)) = √(28*21*4*3) = 84 cm².

Correct Answer: B — 96 cm²

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Q. If the sides of triangle PQR are in the ratio 3:4:5, what type of triangle is it?

A. Equilateral
B. Isosceles
C. Scalene
D. Right

Solution

A triangle with sides in the ratio 3:4:5 is a right triangle.

Correct Answer: D — Right

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Q. If triangle ABC has vertices A(1, 1), B(4, 5), and C(7, 2), what is the length of side AB?

A. 4.0
B. 5.0
C. 3.0
D. 6.0

Solution

Using the distance formula: AB = √((4 - 1)² + (5 - 1)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: B — 5.0

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Q. If triangle ABC has vertices A(1, 2), B(4, 6), and C(1, 6), what is the length of side AB?

A. 5.0
B. 4.0
C. 3.0
D. 6.0

Solution

Using the distance formula: AB = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: B — 4.0

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Q. If triangle ABC has vertices A(1, 2), B(4, 6), and C(7, 2), what is the area of triangle ABC?

A. 12
B. 10
C. 14
D. 8

Solution

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 0.5 * |1(6 - 2) + 4(2 - 2) + 7(2 - 6)| = 0.5 * |4 + 0 - 28| = 12.

Correct Answer: B — 10

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Q. If triangle ABC has vertices A(1, 2), B(4, 6), and C(7, 2), what is the length of side AB?

A. 5.0
B. 4.24
C. 3.0
D. 6.0

Solution

Using the distance formula: AB = √((4 - 1)² + (6 - 2)²) = √(9 + 16) = √25 = 5.0.

Correct Answer: B — 4.24

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Q. If triangle ABC is congruent to triangle DEF, which of the following is true?

A. AB = DE
B. AC = DF
C. BC = EF
D. All of the above

Solution

If two triangles are congruent, all corresponding sides and angles are equal.

Correct Answer: D — All of the above

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Q. If triangle ABC is congruent to triangle DEF, which of the following must be true?

A. AB = DE
B. Angle A = Angle D
C. AC = DF
D. All of the above

Solution

If two triangles are congruent, all corresponding sides and angles are equal.

Correct Answer: D — All of the above

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Q. If triangle ABC is congruent to triangle DEF, which of the following statements is true?

A. AB = DE
B. AC = DF
C. BC = EF
D. All of the above

Solution

All corresponding sides and angles are equal in congruent triangles.

Correct Answer: D — All of the above

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Q. If triangle ABC is isosceles with AB = AC and angle A = 40 degrees, what is the measure of angles B and C?

A. 70 degrees each
B. 80 degrees each
C. 60 degrees each
D. 50 degrees each

Solution

In an isosceles triangle, the base angles are equal. Therefore, angle B = angle C = (180 - 40) / 2 = 70 degrees.

Correct Answer: B — 80 degrees each

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Q. If triangle ABC is isosceles with AB = AC, and angle A = 40 degrees, what are the measures of angles B and C?

A. 70 degrees each
B. 80 degrees each
C. 60 degrees each
D. 40 degrees each

Solution

In an isosceles triangle, angles B and C are equal. Therefore, angle B + angle C = 180 - 40 = 140 degrees, so each is 70 degrees.

Correct Answer: B — 80 degrees each

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Q. If triangle ABC is isosceles with AB = AC, which of the following is true?

A. Angle B = Angle C
B. Angle A = Angle B
C. Angle A = Angle C
D. All angles are equal

Solution

In an isosceles triangle, the angles opposite the equal sides are also equal.

Correct Answer: A — Angle B = Angle C

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Q. If triangle ABC is similar to triangle DEF and the ratio of their corresponding sides is 3:5, what is the ratio of their areas?

A. 3:5
B. 9:25
C. 15:25
D. 5:3

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (3/5)^2 = 9/25.

Correct Answer: B — 9:25

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Q. If triangle ABC is similar to triangle DEF with a scale factor of 2, and the area of triangle ABC is 50 cm², what is the area of triangle DEF?

A. 100 cm²
B. 200 cm²
C. 150 cm²
D. 250 cm²

Solution

The area of similar triangles is proportional to the square of the scale factor. Area of DEF = 50 * (2^2) = 200 cm².

Correct Answer: B — 200 cm²

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Q. If triangle ABC is similar to triangle DEF, and the lengths of sides AB and DE are 6 cm and 9 cm respectively, what is the ratio of their areas?

A. 2:3
B. 3:2
C. 4:9
D. 9:4

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (6/9)² = (2/3)² = 4/9.

Correct Answer: C — 4:9

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Q. If triangle ABC is similar to triangle DEF, and the lengths of sides AB and DE are 4 cm and 8 cm respectively, what is the ratio of the areas of the triangles?

A. 1:2
B. 1:4
C. 2:1
D. 4:1

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (4/8)^2 = (1/2)^2 = 1/4.

Correct Answer: B — 1:4

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Q. If triangle ABC is similar to triangle DEF, and the lengths of sides AB and DE are 4 cm and 8 cm respectively, what is the ratio of their areas?

A. 1:2
B. 1:4
C. 2:1
D. 4:1

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (4/8)² = (1/2)² = 1/4.

Correct Answer: B — 1:4

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Q. If triangle ABC is similar to triangle DEF, and the lengths of sides AB and DE are 6 cm and 9 cm respectively, what is the ratio of the areas of the triangles?

A. 2:3
B. 3:2
C. 4:9
D. 9:4

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (6/9)² = (2/3)² = 4/9.

Correct Answer: C — 4:9

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Q. If triangle ABC is similar to triangle DEF, and the ratio of their corresponding sides is 2:3, what is the ratio of their areas?

A. 2:3
B. 4:9
C. 3:2
D. 9:4

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (2/3)² = 4/9.

Correct Answer: B — 4:9

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Q. If triangle DEF is similar to triangle GHI, and the length of DE is 5 cm and GH is 10 cm, what is the ratio of DE to GH?

A. 1:2
B. 2:1
C. 1:1
D. 5:10

Solution

The ratio of DE to GH is 5:10, which simplifies to 1:2.

Correct Answer: A — 1:2

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Q. If triangle DEF is similar to triangle GHI, and the lengths of DE and GH are 4 cm and 8 cm respectively, what is the ratio of the areas of the two triangles?

A. 1:2
B. 1:4
C. 1:8
D. 1:16

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (4/8)^2 = (1/2)^2 = 1/4.

Correct Answer: B — 1:4

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Q. If triangle DEF is similar to triangle GHI, and the lengths of DE and GH are 4 cm and 8 cm respectively, what is the ratio of their areas?

A. 1:2
B. 1:4
C. 2:1
D. 4:1

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (4/8)^2 = 1/4.

Correct Answer: B — 1:4

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Q. If triangle DEF is similar to triangle XYZ and the length of DE is 4 cm and XY is 8 cm, what is the ratio of DE to XY?

A. 1:2
B. 2:1
C. 1:4
D. 4:1

Solution

The ratio of DE to XY is 4:8, which simplifies to 1:2.

Correct Answer: A — 1:2

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Q. If triangle DEF is similar to triangle XYZ and the length of DE is 4 cm and XY is 8 cm, what is the ratio of their areas?

A. 1:2
B. 1:4
C. 1:8
D. 1:16

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. (4/8)² = 1/4.

Correct Answer: B — 1:4

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Geometry

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