Geometry Study Resources for Competitive Exams

Geometry

Q. In triangle STU, if angle S = 30 degrees and angle T = 70 degrees, what is the length of side SU if ST = 10 cm and TU = 15 cm?

A. 7.5 cm
B. 10 cm
C. 12.5 cm
D. 15 cm

Solution

Using the Law of Sines: SU / sin(80) = 10 / sin(30). Therefore, SU = 10 * sin(80) / sin(30) = 10 * 0.9848 / 0.5 = 19.696 cm.

Correct Answer: C — 12.5 cm

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Q. In triangle STU, if angle S = 45 degrees and angle T = 45 degrees, what is the type of triangle STU?

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

Solution

Since two angles are equal, triangle STU is isosceles.

Correct Answer: D — Isosceles

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Q. In triangle STU, if ST = 12 cm, TU = 16 cm, and SU = 20 cm, what is the perimeter of triangle STU?

A. 28 cm
B. 36 cm
C. 40 cm
D. 48 cm

Solution

The perimeter of triangle STU is ST + TU + SU = 12 + 16 + 20 = 48 cm.

Correct Answer: B — 36 cm

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Q. In triangle STU, if ST = 7 cm, TU = 24 cm, and SU = 25 cm, is triangle STU a right triangle?

A. Yes
B. No
C. Only if angle S is 90 degrees
D. Only if angle T is 90 degrees

Solution

Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625, which equals 25^2. Therefore, triangle STU is a right triangle.

Correct Answer: A — Yes

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Q. In triangle UVW, if angle U = 45 degrees and angle V = 45 degrees, what type of triangle is it?

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

Solution

Since two angles are equal (45 degrees each), triangle UVW is an isosceles triangle.

Correct Answer: B — Isosceles

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Q. In triangle XYZ, if angle X = 45 degrees and angle Y = 45 degrees, what type of triangle is it?

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

Solution

Since two angles are equal, triangle XYZ is an isosceles triangle.

Correct Answer: B — Isosceles

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Q. In triangle XYZ, if angle X = 90 degrees, angle Y = 45 degrees, and side XY = 10 units, what is the length of side XZ?

A. 10√2 units
B. 5√2 units
C. 10 units
D. 5 units

Solution

In a 45-45-90 triangle, the sides opposite the 45-degree angles are equal, and the hypotenuse is √2 times the length of each leg. Thus, XZ = XY√2 = 10√2 units.

Correct Answer: A — 10√2 units

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Q. In triangle XYZ, if angle X is 30 degrees and angle Y is 60 degrees, what is angle Z?

A. 30 degrees
B. 60 degrees
C. 90 degrees
D. 120 degrees

Solution

The sum of the angles in a triangle is 180 degrees. Therefore, angle Z = 180 - (30 + 60) = 90 degrees.

Correct Answer: C — 90 degrees

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Q. In triangle XYZ, if XY = 5 cm, YZ = 12 cm, and XZ = 13 cm, is triangle XYZ a right triangle?

A. Yes
B. No
C. Not enough information
D. Only if XY is the longest side

Solution

Triangle XYZ is a right triangle because 5² + 12² = 25 + 144 = 169 = 13².

Correct Answer: A — Yes

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Q. In triangle XYZ, if XY = 7 cm, YZ = 24 cm, and XZ = 25 cm, is triangle XYZ a right triangle?

A. Yes
B. No
C. Only if angle Y is 90 degrees
D. Only if angle Z is 90 degrees

Solution

Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2, thus triangle XYZ is a right triangle.

Correct Answer: A — Yes

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Q. In triangle XYZ, if XY = 8 cm, XZ = 6 cm, and YZ = 10 cm, what type of triangle is it?

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

Solution

Triangle XYZ is scalene because all sides have different lengths.

Correct Answer: C — Scalene

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Q. In triangle XYZ, if XY = 8 cm, XZ = 6 cm, and YZ = 10 cm, which side is the longest?

A. XY
B. XZ
C. YZ
D. All sides are equal

Solution

YZ is the longest side since 10 cm is greater than both 8 cm and 6 cm.

Correct Answer: C — YZ

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Q. In triangle XYZ, if XY = 8 cm, YZ = 6 cm, and XZ = 10 cm, what type of triangle is it?

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

Solution

Triangle XYZ has all sides of different lengths (8 cm, 6 cm, 10 cm), so it is a scalene triangle.

Correct Answer: C — Scalene

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Q. In triangle XYZ, if XY = 8, YZ = 6, and XZ = 10, what type of triangle is it?

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

Solution

Triangle XYZ is a right triangle because 8^2 + 6^2 = 10^2.

Correct Answer: D — Right

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Q. In triangle XYZ, if XY = 8, YZ = 6, and XZ = 10, which side is the longest?

A. XY
B. YZ
C. XZ
D. All sides are equal

Solution

The longest side in triangle XYZ is XZ, which measures 10 units.

Correct Answer: C — XZ

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Q. Triangle DEF is similar to triangle XYZ. If the lengths of DE and XY are 4 cm and 8 cm respectively, what is the ratio of the areas of the 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 = 1/4.

Correct Answer: B — 1:4

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Q. Triangle DEF is similar to triangle XYZ. If the sides of triangle DEF are 3 cm, 4 cm, and 5 cm, what is the length of the longest side of triangle XYZ if its shortest side is 6 cm?

A. 8 cm
B. 9 cm
C. 10 cm
D. 12 cm

Solution

The ratio of the sides of similar triangles is constant. The shortest side ratio is 6/3 = 2. Therefore, the longest side of triangle XYZ = 5 * 2 = 10 cm.

Correct Answer: C — 10 cm

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Q. Triangle GHI is an isosceles triangle with GH = GI and angle H = 30 degrees. What is the measure of angle I?

A. 30 degrees
B. 60 degrees
C. 75 degrees
D. 90 degrees

Solution

In an isosceles triangle, the base angles are equal. Therefore, angle G + angle I = 150 degrees. Each angle = 60 degrees.

Correct Answer: B — 60 degrees

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Q. Two chords AB and CD of a circle intersect at point E. If AE = 3 cm, EB = 5 cm, and CE = 4 cm, what is the length of ED?

A. 6 cm
B. 8 cm
C. 5 cm
D. 7 cm

Solution

Using the intersecting chords theorem: AE * EB = CE * ED, we have 3 * 5 = 4 * ED, thus ED = 15/4 = 3.75 cm.

Correct Answer: B — 8 cm

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Q. Two circles are tangent to each other. If the radius of the first circle is 3 cm and the second is 5 cm, what is the distance between their centers?

A. 2 cm
B. 8 cm
C. 3 cm
D. 5 cm

Solution

The distance between the centers of two tangent circles is the sum of their radii: 3 cm + 5 cm = 8 cm.

Correct Answer: B — 8 cm

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Q. Two circles have radii of 3 cm and 5 cm. What is the distance between their centers if they are externally tangent?

A. 2 cm
B. 8 cm
C. 5 cm
D. 3 cm

Solution

Distance = r1 + r2 = 3 + 5 = 8 cm.

Correct Answer: B — 8 cm

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Q. Two circles intersect at points A and B. If the angle ∠AOB is 60°, what is the measure of the angle ∠APB where P is any point on the circumference of the circles?

A. 30°
B. 60°
C. 90°
D. 120°

Solution

Angle ∠APB = 1/2 ∠AOB = 1/2(60°) = 30°.

Correct Answer: A — 30°

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Q. Two circles intersect at points A and B. If the angle ∠APB is 60°, what is the measure of the angle ∠AOB, where O is the center of the circle?

A. 30°
B. 60°
C. 120°
D. 90°

Solution

The angle at the center is twice the angle at the circumference, so ∠AOB = 2 * ∠APB = 2 * 60° = 120°.

Correct Answer: C — 120°

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Q. Two circles intersect at points A and B. If the line segment AB is the common chord, what can be said about the perpendicular from the center of either circle to AB?

A. It bisects AB
B. It is equal to AB
C. It is longer than AB
D. It is shorter than AB

Solution

The perpendicular from the center of a circle to a chord bisects the chord.

Correct Answer: A — It bisects AB

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Q. Two circles intersect at points A and B. If the radius of the first circle is 4 cm and the second is 6 cm, what is the maximum distance between the centers of the circles?

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

Solution

Maximum distance = r1 + r2 = 4 + 6 = 10 cm.

Correct Answer: A — 10 cm

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Q. Two circles intersect at points A and B. If the radius of the first circle is 4 cm and the second circle is 6 cm, what is the maximum distance between the centers of the circles?

A. 10 cm
B. 8 cm
C. 2 cm
D. 12 cm

Solution

Maximum distance = r1 + r2 = 4 + 6 = 10 cm.

Correct Answer: A — 10 cm

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Q. Two circles intersect at points A and B. If the radius of the first circle is 5 cm and the second is 3 cm, what is the maximum distance between the centers of the circles?

A. 8 cm
B. 10 cm
C. 6 cm
D. 7 cm

Solution

Maximum distance = r1 + r2 = 5 + 3 = 8 cm.

Correct Answer: A — 8 cm

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Q. Two circles intersect at points A and B. What is the relationship between the angles ∠AOB and ∠APB, where O is the center of one circle and P is the center of the other?

A. ∠AOB = ∠APB
B. ∠AOB = 2∠APB
C. ∠AOB = ½∠APB
D. ∠AOB + ∠APB = 180 degrees

Solution

The angle ∠AOB is twice the angle ∠APB because of the inscribed angle theorem, which states that the angle at the center is twice the angle at the circumference.

Correct Answer: B — ∠AOB = 2∠APB

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Q. Two lines are parallel, and a transversal intersects them, creating angles of 75 degrees and x degrees. What is the value of x?

A. 75 degrees
B. 105 degrees
C. 180 degrees
D. 90 degrees

Solution

The angles are supplementary, so x = 180 - 75 = 105 degrees.

Correct Answer: B — 105 degrees

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Q. Two parallel lines are cut by a transversal, creating angles of 120 degrees and x degrees. What is the value of x?

A. 60 degrees
B. 120 degrees
C. 180 degrees
D. 30 degrees

Solution

The angles formed are supplementary. Therefore, 120 + x = 180, which gives x = 60 degrees.

Correct Answer: A — 60 degrees

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Geometry

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