Trigonometry Study Resources for Competitive Exams

Trigonometry

Heights & Distances Properties of Triangles Trigonometric Equations Trigonometric Ratios & Identities

Q. If the sides of triangle ABC are 7 cm, 24 cm, and 25 cm, what is the perimeter of the triangle?

A. 50 cm
B. 55 cm
C. 60 cm
D. 65 cm

Solution

Perimeter = a + b + c = 7 + 24 + 25 = 56 cm.

Correct Answer: A — 50 cm

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

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

Solution

A triangle with sides in the ratio 3:4:5 is a right triangle, as it satisfies the Pythagorean theorem.

Correct Answer: C — Right

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Q. In triangle ABC, if a = 7, b = 24, and c = 25, what is the area of the triangle?

A. 84
B. 96
C. 120
D. 144

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.

Correct Answer: A — 84

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Q. In triangle ABC, if AB = 10 cm, AC = 6 cm, and angle A = 30°, what is the length of side BC?

A. 8 cm
B. 7 cm
C. 5 cm
D. 4 cm

Solution

Using the Law of Cosines: BC² = AB² + AC² - 2 * AB * AC * cos(A) = 10² + 6² - 2 * 10 * 6 * (√3/2) = 100 + 36 - 60√3. BC = √(100 + 36 - 60√3) ≈ 7 cm.

Correct Answer: B — 7 cm

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Q. In triangle ABC, if AB = 7 cm, AC = 24 cm, and BC = 25 cm, is triangle ABC a right triangle?

A. Yes
B. No
C. Cannot be determined
D. Only if angle A is 90°

Solution

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

Correct Answer: A — Yes

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Q. In triangle ABC, if AB = 7 cm, AC = 24 cm, and BC = 25 cm, what is the area of the triangle?

A. 84 cm²
B. 96 cm²
C. 120 cm²
D. 140 cm²

Solution

Using Heron's formula, the semi-perimeter 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. In triangle ABC, if AB = 8, AC = 6, and BC = 10, what is the semi-perimeter?

A. 12
B. 14
C. 16
D. 18

Solution

Semi-perimeter s = (AB + AC + BC) / 2 = (8 + 6 + 10) / 2 = 12.

Correct Answer: B — 14

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Q. In triangle ABC, if angle A = 45 degrees and angle B = 45 degrees, what is angle C?

A. 45 degrees
B. 60 degrees
C. 90 degrees
D. 135 degrees

Solution

The sum of angles in a triangle is 180 degrees. Therefore, angle C = 180 - (45 + 45) = 90 degrees.

Correct Answer: C — 90 degrees

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Q. In triangle ABC, if angle A = 45 degrees and angle B = 45 degrees, what is the relationship between sides a, b, and c?

A. a = b
B. a > b
C. a < b
D. a + b = c

Solution

In an isosceles triangle with angles A and B equal, the sides opposite those angles are equal, hence a = b.

Correct Answer: A — a = b

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Q. In triangle ABC, if angle A = 45 degrees and angle B = 45 degrees, what is the type of triangle?

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

Solution

Since two angles are equal, triangle ABC is isosceles.

Correct Answer: B — Isosceles

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Q. In triangle ABC, if angle A = 45 degrees and side a = 10 cm, what is the length of side b if angle B = 60 degrees?

A. 8.66 cm
B. 10 cm
C. 12.25 cm
D. 15 cm

Solution

Using the Law of Sines: a/sin(A) = b/sin(B). Therefore, b = a * (sin(B)/sin(A)) = 10 * (sin(60)/sin(45)) = 10 * (√3/2)/(√2/2) = 10 * √3/√2 = 10 * √(3/2) = 8.66 cm.

Correct Answer: A — 8.66 cm

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Q. In triangle ABC, if angle A = 45° and angle B = 45°, what is angle C?

A. 45°
B. 60°
C. 75°
D. 90°

Solution

Angle C = 180° - (angle A + angle B) = 180° - (45° + 45°) = 90°.

Correct Answer: D — 90°

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Q. In triangle ABC, if angle A = 45° and side a = 10, what is the length of side b if angle B = 60°?

A. 8.66
B. 7.5
C. 5
D. 10

Solution

Using the Law of Sines: b/a = sin(B)/sin(A) => b = a * (sin(B)/sin(A)) = 10 * (√3/2)/(√2/2) = 10 * √3/√2 = 10 * 8.66/10 = 8.66.

Correct Answer: A — 8.66

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Q. In triangle ABC, if angle A = 60 degrees and angle B = 70 degrees, what is angle C?

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

Solution

Angle C = 180 - (angle A + angle B) = 180 - (60 + 70) = 50 degrees.

Correct Answer: A — 50 degrees

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Q. In triangle ABC, if angle A = 60 degrees and angle B = 70 degrees, what is the measure of angle C?

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

Solution

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

Correct Answer: A — 50 degrees

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Q. In triangle ABC, if angle A = 60° and angle B = 70°, what is angle C?

A. 50°
B. 60°
C. 70°
D. 80°

Solution

Angle C = 180° - (angle A + angle B) = 180° - (60° + 70°) = 50°.

Correct Answer: A — 50°

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Q. In triangle ABC, if the angles are in the ratio 2:3:4, what is the measure of the largest angle?

A. 60 degrees
B. 80 degrees
C. 90 degrees
D. 120 degrees

Solution

Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 => 9x = 180 => x = 20. The largest angle = 4x = 80 degrees.

Correct Answer: B — 80 degrees

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Q. In triangle ABC, if the coordinates of A, B, and C are (1, 2), (4, 6), and (7, 2) respectively, what is the length of side AB?

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

Solution

Length of AB = √[(4-1)² + (6-2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.

Correct Answer: B — 5

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Q. In triangle ABC, if the coordinates of A, B, and C are (1, 2), (4, 6), and (7, 2) respectively, what is the area of triangle ABC?

A. 12
B. 14
C. 16
D. 18

Solution

Using the formula for the area of a triangle given vertices, Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 1/2 | 1(6-2) + 4(2-2) + 7(2-6) | = 1/2 | 4 + 0 - 28 | = 12.

Correct Answer: A — 12

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Q. In triangle ABC, if the lengths of sides a = 10, b = 24, and angle C = 60 degrees, find the length of side c.

A. 20
B. 22
C. 24
D. 26

Solution

Using the cosine rule: c^2 = a^2 + b^2 - 2ab*cos(C) = 10^2 + 24^2 - 2*10*24*(1/2) = 100 + 576 - 240 = 436. Thus, c = √436 = 20.

Correct Answer: A — 20

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Q. In triangle ABC, if the lengths of sides a, b, and c are 5, 12, and 13 respectively, what is the perimeter of the triangle?

A. 30
B. 25
C. 20
D. 18

Solution

Perimeter = a + b + c = 5 + 12 + 13 = 30.

Correct Answer: B — 25

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Q. In triangle ABC, if the lengths of sides a, b, and c are 7, 24, and 25 respectively, what type of triangle is it?

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

Solution

Since 7² + 24² = 49 + 576 = 625 = 25², triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the lengths of sides a, b, and c are 7, 24, and 25 respectively, what is the area of the triangle?

A. 84
B. 96
C. 120
D. 168

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.

Correct Answer: B — 96

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Q. In triangle ABC, if the lengths of sides a, b, and c are 8, 15, and 17 respectively, what is the type of triangle?

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

Solution

Since 8² + 15² = 17², triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the lengths of sides are 8 cm, 15 cm, and 17 cm, what is the type of triangle?

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

Solution

Since 8² + 15² = 64 + 225 = 289 = 17², triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the lengths of the sides are 10 cm, 24 cm, and 26 cm, what is the type of triangle?

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

Solution

Since 10² + 24² = 100 + 576 = 676 = 26², triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the lengths of the sides are 5, 12, and 13, what is the perimeter?

A. 30
B. 25
C. 20
D. 18

Solution

Perimeter = a + b + c = 5 + 12 + 13 = 30.

Correct Answer: B — 25

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Q. In triangle ABC, if the lengths of the sides are 8 cm, 15 cm, and 17 cm, what is the perimeter?

A. 30 cm
B. 40 cm
C. 50 cm
D. 60 cm

Solution

Perimeter = 8 + 15 + 17 = 40 cm.

Correct Answer: A — 30 cm

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Q. In triangle ABC, if the lengths of the sides are 8 cm, 15 cm, and 17 cm, what is the type of triangle?

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

Solution

Since 8² + 15² = 64 + 225 = 289 = 17², triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the lengths of the sides are 8, 15, and 17, what is the area of the triangle?

A. 60
B. 80
C. 120
D. 150

Solution

Using Heron's formula, the semi-perimeter s = (8 + 15 + 17)/2 = 20. Area = √[s(s-a)(s-b)(s-c)] = √[20(20-8)(20-15)(20-17)] = √[20*12*5*3] = 60.

Correct Answer: A — 60

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Trigonometry

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