Geometry Study Resources for Competitive Exams

Geometry

Q. What is the radius of a circle if the circumference is 31.4 cm?

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Solution

The circumference C = 2πr. Thus, r = C/(2π) = 31.4/(2π) ≈ 5 cm.

Correct Answer: B — 10 cm

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Q. What is the radius of a circle inscribed in a triangle with sides 7 cm, 8 cm, and 9 cm?

A. 4 cm
B. 3 cm
C. 5 cm
D. 6 cm

Solution

Semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. Area = √[s(s-a)(s-b)(s-c)] = √[12(12-7)(12-8)(12-9)] = √[12*5*4*3] = 12√5 cm². Radius = Area/s = 12√5/12 = √5 cm.

Correct Answer: B — 3 cm

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Q. What is the radius of a circle whose area is 113.04 cm²?

A. 6 cm
B. 7 cm
C. 8 cm
D. 9 cm

Solution

Area = πr², so r = √(Area/π) = √(113.04/π) ≈ 6 cm.

Correct Answer: B — 7 cm

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Q. What is the radius of a circle whose area is 50π cm²?

A. 5 cm
B. 10 cm
C. 7.07 cm
D. 12.56 cm

Solution

Area = πr², so r² = 50, r = √50 = 7.07 cm.

Correct Answer: C — 7.07 cm

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Q. What is the radius of a circle with a circumference of 31.4 units?

A. 5 units
B. 10 units
C. 15 units
D. 20 units

Solution

The formula for circumference is C = 2πr. Solving for r gives r = C/(2π) = 31.4/(2π) = 5 units.

Correct Answer: A — 5 units

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Q. What is the radius of a circle with the equation (x - 2)² + (y + 3)² = 16?

A. 4
B. 8
C. 2
D. 3

Solution

The standard form of a circle is (x - h)² + (y - k)² = r². Here, r² = 16, so r = √16 = 4.

Correct Answer: A — 4

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Q. What is the radius of a circle with the equation (x - 2)² + (y + 3)² = 25?

A. 5
B. 10
C. 25
D. 20

Solution

Radius = √25 = 5.

Correct Answer: A — 5

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Q. What is the radius of a circle with the equation (x - 3)² + (y + 2)² = 25?

A. 5
B. 10
C. 15
D. 20

Solution

The standard form of a circle is (x - h)² + (y - k)² = r². Here, r² = 25, so r = √25 = 5.

Correct Answer: A — 5

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Q. What is the radius of a circle with the equation (x - 4)² + (y + 2)² = 36?

A. 6
B. 4
C. 8
D. 5

Solution

The standard form of a circle is (x - h)² + (y - k)² = r². Here, r² = 36, so r = √36 = 6.

Correct Answer: A — 6

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Q. What is the ratio in which the point (3, 4) divides the line segment joining (1, 2) and (5, 6)?

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

Solution

Using the section formula, we can find the ratio by solving for m:n in the equations derived from the coordinates.

Correct Answer: B — 2:1

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Q. What is the ratio in which the point (3, 5) divides the line segment joining (1, 2) and (5, 8)?

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

Solution

Using the section formula, we find the ratio is 2:1.

Correct Answer: B — 2:1

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Q. What is the ratio of the areas of two similar triangles if the ratio of their corresponding sides is 3:4?

A. 3:4
B. 9:16
C. 12:16
D. 1:1

Solution

The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides. Therefore, (3/4)^2 = 9/16.

Correct Answer: B — 9:16

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Q. What is the ratio of the areas of two similar triangles if the ratio of their corresponding sides is 3:5?

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. (3/5)^2 = 9/25.

Correct Answer: B — 9:25

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Q. What is the relationship between the angles formed by two intersecting lines?

A. They are equal.
B. They are complementary.
C. They are supplementary.
D. They are not related.

Solution

The angles formed by two intersecting lines are supplementary, meaning they add up to 180 degrees.

Correct Answer: C — They are supplementary.

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Q. What is the relationship between the angles formed by two parallel lines cut by a transversal?

A. All angles are equal.
B. Corresponding angles are equal.
C. Alternate exterior angles are equal.
D. Both B and C.

Solution

Corresponding angles and alternate exterior angles are equal when two parallel lines are cut by a transversal.

Correct Answer: D — Both B and C.

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Q. What is the relationship between the angles formed by two tangents drawn from a point outside a circle?

A. They are equal
B. They are supplementary
C. They are complementary
D. They are congruent

Solution

The angles formed by two tangents drawn from a point outside a circle to the points of tangency are equal.

Correct Answer: A — They are equal

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Q. What is the relationship between the angles formed by two tangents drawn from a point outside the circle?

A. They are equal
B. They are supplementary
C. They are complementary
D. They are congruent

Solution

The angles formed by two tangents drawn from a point outside the circle are equal.

Correct Answer: A — They are equal

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Q. What is the relationship between the angles in similar triangles?

A. They are equal
B. They are supplementary
C. They are complementary
D. They are congruent

Solution

In similar triangles, corresponding angles are equal.

Correct Answer: A — They are equal

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Q. What is the relationship between the angles of a triangle and its sides?

A. Larger angles are opposite longer sides
B. Smaller angles are opposite longer sides
C. All angles are equal
D. None of the above

Solution

In a triangle, larger angles are always opposite longer sides, which is a fundamental property of triangles.

Correct Answer: A — Larger angles are opposite longer sides

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Q. What is the relationship between the angles of two congruent triangles?

A. They are always equal
B. They can be different
C. They are supplementary
D. They are complementary

Solution

In congruent triangles, all corresponding angles are equal.

Correct Answer: A — They are always equal

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Q. What is the relationship between the angles of two parallel lines cut by a transversal?

A. They are all equal
B. They are supplementary
C. They are complementary
D. They are congruent

Solution

When two parallel lines are cut by a transversal, the interior angles on the same side are supplementary.

Correct Answer: B — They are supplementary

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Q. What is the relationship between the angles subtended by an arc at the center and at any point on the remaining part of the circle?

A. The angle at the center is half
B. The angle at the center is double
C. They are equal
D. The angle at the center is zero

Solution

The angle subtended by an arc at the center is always double the angle subtended at any point on the remaining part of the circle.

Correct Answer: B — The angle at the center is double

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Q. What is the relationship between the angles subtended by the same arc at the center and at any point on the remaining part of the circle?

A. They are equal
B. The angle at the center is double
C. The angle at the center is half
D. They are supplementary

Solution

The angle subtended at the center is always double the angle subtended at any point on the remaining part of the circle.

Correct Answer: B — The angle at the center is double

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Q. What is the relationship between the angles subtended by the same arc at the center and at the circumference?

A. They are equal
B. The angle at the center is twice the angle at the circumference
C. The angle at the circumference is twice the angle at the center
D. They are complementary

Solution

The angle subtended at the center is always twice the angle subtended at the circumference by the same arc.

Correct Answer: B — The angle at the center is twice the angle at the circumference

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Q. What is the relationship between the angles subtended by the same arc at the center and at any point on the circumference?

A. The angle at the center is double
B. The angle at the center is half
C. They are equal
D. They are supplementary

Solution

The angle subtended at the center is always double the angle subtended at any point on the circumference.

Correct Answer: A — The angle at the center is double

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Q. What is the relationship between the angles subtended by the same arc at the center and on the circumference of a circle?

A. They are equal
B. The angle at the center is twice the angle at the circumference
C. The angle at the circumference is twice the angle at the center
D. They are complementary

Solution

The angle subtended at the center of a circle is always twice the angle subtended at the circumference by the same arc.

Correct Answer: B — The angle at the center is twice the angle at the circumference

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Q. What is the relationship between the angles subtended by the same arc at the center and at the circumference of the circle?

A. The angle at the center is half of the angle at the circumference
B. The angle at the center is equal to the angle at the circumference
C. The angle at the center is double the angle at the circumference
D. There is no relationship

Solution

The angle subtended at the center of the circle is always double the angle subtended at the circumference by the same arc.

Correct Answer: C — The angle at the center is double the angle at the circumference

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Q. What is the relationship between the consecutive interior angles formed by a transversal intersecting two parallel lines?

A. They are equal.
B. They are complementary.
C. They are supplementary.
D. They are adjacent.

Solution

Consecutive interior angles are supplementary when two parallel lines are cut by a transversal.

Correct Answer: C — They are supplementary.

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Q. What is the relationship between the exterior angle and the interior angle on the same side of a transversal intersecting two parallel lines?

A. They are equal.
B. They are complementary.
C. They are supplementary.
D. They are not related.

Solution

The Exterior Angle Theorem states that the exterior angle is supplementary to the interior angle on the same side of the transversal.

Correct Answer: C — They are supplementary.

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Q. What is the relationship between the exterior angle and the interior angle on the same side when two parallel lines are cut by a transversal?

A. They are equal.
B. They are complementary.
C. They are supplementary.
D. They are not related.

Solution

The exterior angle and the interior angle on the same side are supplementary when two parallel lines are cut by a transversal.

Correct Answer: C — They are supplementary.

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Geometry

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