Basic Geometric Concepts - Exam Prep for Students

Basic Geometric Concepts - Proof-based Questions

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

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

Q. If two lines are parallel, what can be said about the alternate interior angles formed by a transversal?

Solution

When two parallel lines are cut by a transversal, the alternate interior angles are equal.

Correct Answer: A — They are equal

Q. If two triangles have sides in the ratio 3:4:5, what can be said about their similarity?

Solution

Triangles with sides in the same ratio are similar.

Correct Answer: A — They are similar

Q. In triangle DEF, if DE = 8 cm, EF = 6 cm, and DF = 10 cm, is triangle DEF a right triangle?

Solution

Using the Pythagorean theorem: 8² + 6² = 64 + 36 = 100 = 10², so triangle DEF is a right triangle.

Correct Answer: A — Yes

Q. What is the length of the diagonal of a rectangle with a length of 12 cm and a width of 9 cm?

Solution

Using the Pythagorean theorem: d² = length² + width² = 12² + 9² = 144 + 81 = 225, so d = 15 cm.

Correct Answer: A — 15 cm

Q. What is the length of the diagonal of a rectangle with length 12 cm and width 9 cm?

Solution

Using the Pythagorean theorem: d² = length² + width² = 12² + 9² = 144 + 81 = 225, so d = 15 cm.

Correct Answer: A — 15 cm

Q. What is the perimeter of a rectangle with a length of 10 cm and a width of 5 cm?

Solution

Perimeter = 2(length + width) = 2(10 + 5) = 30 cm.

Correct Answer: A — 30 cm

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