Geometry Study Resources for Competitive Exams

Geometry

Q. If a tangent from point P touches a circle at point T, and the radius OT is 5 cm, what is the length of PT if PT is perpendicular to OT?

A. 5√2 cm
B. 5 cm
C. 10 cm
D. 25 cm

Solution

Since PT is perpendicular to OT, PT is equal to the radius OT, which is 5 cm.

Correct Answer: B — 5 cm

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Q. If a tangent from point P touches a circle at point T, what is the relationship between the radius OT and the tangent PT?

A. OT is equal to PT
B. OT is perpendicular to PT
C. OT is parallel to PT
D. OT is longer than PT

Solution

The radius drawn to the point of tangency is perpendicular to the tangent line at that point.

Correct Answer: B — OT is perpendicular to PT

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Q. If a tangent from point P touches the circle at point T, and the radius OT is 6 cm, what is the length of PT if PT is the tangent?

A. 6 cm
B. 12 cm
C. 8 cm
D. 10 cm

Solution

The length of the tangent PT from point P to the point of tangency T is equal to the radius OT, which is 6 cm.

Correct Answer: A — 6 cm

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Q. If a tangent from point P touches the circle at point T, which of the following statements is true?

A. PT is perpendicular to the radius OT
B. PT is parallel to the radius OT
C. PT is equal to the radius OT
D. PT bisects the angle OTP

Solution

The tangent to a circle at any point is perpendicular to the radius drawn to the point of tangency.

Correct Answer: A — PT is perpendicular to the radius OT

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Q. If a tangent is drawn to a circle from a point outside the circle, what is the relationship between the radius and the tangent at the point of contact?

A. They are equal
B. They are perpendicular
C. They are parallel
D. They are collinear

Solution

The tangent to a circle at any point is perpendicular to the radius drawn to the point of contact.

Correct Answer: B — They are perpendicular

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Q. If a tangent touches a circle at point A and a line from the center to point A is drawn, what is the angle between the tangent and the radius?

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

The tangent to a circle at any point is perpendicular to the radius drawn to the point of contact, hence the angle is 90 degrees.

Correct Answer: C — 90 degrees

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Q. If a transversal intersects two parallel lines and creates an angle of 40 degrees, what is the measure of the vertically opposite angle?

A. 40 degrees
B. 80 degrees
C. 60 degrees
D. 20 degrees

Solution

Vertically opposite angles are equal, so the angle is also 40 degrees.

Correct Answer: A — 40 degrees

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Q. If a transversal intersects two parallel lines and forms a pair of corresponding angles, what can be said about their measures?

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

Solution

Corresponding angles are equal when a transversal intersects two parallel lines.

Correct Answer: A — They are equal.

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Q. If a transversal intersects two parallel lines and forms an angle of 30 degrees, what is the measure of the corresponding angle on the other line?

A. 30 degrees
B. 60 degrees
C. 90 degrees
D. 150 degrees

Solution

Corresponding angles are equal when a transversal intersects two parallel lines.

Correct Answer: A — 30 degrees

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Q. If a transversal intersects two parallel lines, what is the sum of the interior angles on the same side of the transversal?

A. 90 degrees
B. 180 degrees
C. 360 degrees
D. It varies.

Solution

The interior angles on the same side of the transversal are supplementary, summing to 180 degrees.

Correct Answer: B — 180 degrees

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Q. If a trapezoid has bases of lengths 10 cm and 6 cm, and a height of 4 cm, what is its area?

A. 32 cm²
B. 40 cm²
C. 24 cm²
D. 28 cm²

Solution

The area of a trapezoid is calculated using the formula (1/2) × (base1 + base2) × height. Therefore, area = (1/2) × (10 cm + 6 cm) × 4 cm = 32 cm².

Correct Answer: A — 32 cm²

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Q. If a trapezoid has bases of lengths 8 cm and 12 cm, what is the length of the midsegment?

A. 10 cm
B. 8 cm
C. 12 cm
D. 20 cm

Solution

The length of the midsegment of a trapezoid is the average of the lengths of the two bases. Midsegment = (8 + 12) / 2 = 10 cm.

Correct Answer: A — 10 cm

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Q. If a triangle has a base of 15 cm and a height of 10 cm, what is the area?

A. 75 cm²
B. 150 cm²
C. 100 cm²
D. 50 cm²

Solution

Area = 1/2 * base * height = 1/2 * 15 * 10 = 75 cm².

Correct Answer: A — 75 cm²

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Q. If a triangle has an area of 24 cm² and a base of 8 cm, what is the height?

A. 6 cm
B. 8 cm
C. 4 cm
D. 3 cm

Solution

Area = 1/2 * base * height; 24 = 1/2 * 8 * height; height = 6 cm.

Correct Answer: A — 6 cm

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Q. If a triangle has an area of 36 cm² and a base of 9 cm, what is its height?

A. 8 cm
B. 6 cm
C. 4 cm
D. 10 cm

Solution

Area = 1/2 * base * height. 36 = 1/2 * 9 * height. Height = (36 * 2) / 9 = 8 cm.

Correct Answer: B — 6 cm

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Q. If a triangle has angles measuring 50 degrees and 60 degrees, what is the measure of the third angle?

A. 70 degrees
B. 80 degrees
C. 90 degrees
D. 100 degrees

Solution

The sum of the angles in a triangle is always 180 degrees. Therefore, the third angle is 180 - (50 + 60) = 70 degrees.

Correct Answer: A — 70 degrees

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Q. If a triangle has angles of 50 degrees and 60 degrees, what is the measure of the third angle?

A. 70 degrees
B. 80 degrees
C. 90 degrees
D. 100 degrees

Solution

The sum of angles in a triangle is 180 degrees. Therefore, the third angle is 180 - 50 - 60 = 70 degrees.

Correct Answer: A — 70 degrees

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Q. If a triangle has sides of lengths 5, 12, and 13, what type of triangle is it?

A. Acute
B. Obtuse
C. Right
D. Equilateral

Solution

A triangle with sides a, b, and c (where c is the longest side) is a right triangle if a² + b² = c². Here, 5² + 12² = 25 + 144 = 169 = 13², so it is a right triangle.

Correct Answer: C — Right

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Q. If a triangle has sides of lengths 6 cm, 8 cm, and 10 cm, what is the perimeter of the triangle?

A. 24 cm
B. 20 cm
C. 18 cm
D. 22 cm

Solution

The perimeter of a triangle is the sum of its sides: 6 + 8 + 10 = 24 cm.

Correct Answer: A — 24 cm

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Q. If a triangle has sides of lengths 6 cm, 8 cm, and 10 cm, what type of triangle is it?

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

Solution

This triangle is a right triangle because it satisfies the Pythagorean theorem: 6^2 + 8^2 = 10^2 (36 + 64 = 100).

Correct Answer: D — Right

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Q. If a triangle has sides of lengths 7 cm, 24 cm, and 25 cm, what type of triangle is it?

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

Solution

This triangle is a right triangle because 7² + 24² = 25² (49 + 576 = 625).

Correct Answer: D — Right

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Q. If a triangle has vertices at (1, 1), (4, 1), and (1, 5), what is its perimeter?

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

Solution

Perimeter = AB + BC + CA = 3 + 3 + 4 = 10.

Correct Answer: B — 10

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Q. If a triangle has vertices at (1, 2), (4, 6), and (1, 6), what is its area?

A. 10
B. 12
C. 8
D. 6

Solution

Area = 0.5 * |(1(6-6) + 4(6-2) + 1(2-6))| = 0.5 * |0 + 16 - 4| = 6.

Correct Answer: B — 12

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Q. If a triangle has vertices at (1, 2), (4, 6), and (7, 2), what is its perimeter?

A. 18
B. 20
C. 22
D. 16

Solution

Calculate the lengths of the sides using the distance formula: AB = √((4-1)² + (6-2)²) = 5, BC = √((7-4)² + (2-6)²) = 5, CA = √((1-7)² + (2-2)²) = 6. Perimeter = 5 + 5 + 6 = 16.

Correct Answer: A — 18

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Q. If a triangle has vertices at A(0, 0), B(4, 0), and C(0, 3), what is its area?

A. 6
B. 12
C. 8
D. 10

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. If a triangle has vertices at A(0, 0), B(4, 0), and C(0, 3), what is the area of the triangle?

A. 6
B. 12
C. 8
D. 10

Solution

Area = (1/2) * base * height = (1/2) * 4 * 3 = 6.

Correct Answer: A — 6

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Q. If a triangle is inscribed in a circle of radius 10 cm, what is the maximum area of the triangle?

A. 50 cm²
B. 100 cm²
C. 75 cm²
D. 80 cm²

Solution

Maximum area = (1/2) * r² * sin(θ) = (1/2) * 10² * 1 = 50 cm².

Correct Answer: B — 100 cm²

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Q. If a triangle is inscribed in a circle, what is the relationship between the triangle's angles and the circle's angles?

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

Solution

The angles of the triangle inscribed in a circle are supplementary to the angles subtended by the same arc at the center.

Correct Answer: B — They are supplementary

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Q. If angle 1 and angle 2 are alternate exterior angles formed by a transversal intersecting two parallel lines, what is true about their measures?

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

Solution

Alternate exterior angles are equal when two parallel lines are cut by a transversal.

Correct Answer: A — They are equal.

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Q. If angle 1 and angle 2 are alternate interior angles and angle 1 measures 55°, what is the measure of angle 2?

A. 55°
B. 125°
C. 90°
D. 45°

Solution

Alternate interior angles are equal, so angle 2 also measures 55°.

Correct Answer: A — 55°

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Geometry

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