Properties of Triangles for Competitive Exam Preparation

Properties of Triangles

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 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 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 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, 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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Q. In triangle ABC, if the lengths of the sides are 8, 15, and 17, 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 a = 5, b = 12, and c = 13, what is the perimeter of the triangle?

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

Solution

The perimeter of a triangle is the sum of its sides. Therefore, 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 a = 8, b = 15, and c = 17, what is the perimeter?

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

Solution

The perimeter of a triangle is the sum of its sides: a + b + c = 8 + 15 + 17 = 40.

Correct Answer: A — 30

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Q. In triangle ABC, if the lengths of the sides are a = 8, b = 15, and c = 17, what is the value of cos A?

A. 0.5
B. 0.6
C. 0.8
D. 0.9

Solution

Using the cosine rule, cos A = (b² + c² - a²) / (2bc) = (15² + 17² - 8²) / (2 * 15 * 17) = 0.8.

Correct Answer: C — 0.8

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

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

Solution

Since the sides are in the ratio of a Pythagorean triplet (3, 4, 5), triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the sides are in the ratio 3:4:5, what is the nature of the triangle?

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

Solution

The sides satisfy the Pythagorean theorem, hence it is a right triangle.

Correct Answer: C — Right

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Q. In triangle ABC, if the sides 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 MNO, if angle M = 45 degrees and angle N = 45 degrees, what is angle O?

A. 90 degrees
B. 45 degrees
C. 60 degrees
D. 30 degrees

Solution

Angle O = 180 - (angle M + angle N) = 180 - (45 + 45) = 90 degrees.

Correct Answer: A — 90 degrees

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Q. In triangle PQR, if PQ = 10 cm, QR = 24 cm, and PR = 26 cm, what is the area of the triangle?

A. 120 cm²
B. 120√3 cm²
C. 240 cm²
D. 48 cm²

Solution

Using Heron's formula, s = (10 + 24 + 26)/2 = 30. Area = √(30(30-10)(30-24)(30-26)) = √(30*20*6*4) = 120 cm².

Correct Answer: A — 120 cm²

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Q. In triangle XYZ, if XY = 8 cm, YZ = 15 cm, and XZ = 17 cm, is it a right triangle?

A. Yes
B. No
C. Cannot be determined
D. Only if XY is the hypotenuse

Solution

Since 8^2 + 15^2 = 17^2, triangle XYZ is a right triangle.

Correct Answer: A — Yes

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Q. The area of triangle ABC is 24 cm², and the base BC = 8 cm. What is the height from A to BC?

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. The area of triangle ABC is 30 square units, and the base BC is 10 units. What is the height from A to BC?

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

Solution

Area = 1/2 * base * height => 30 = 1/2 * 10 * height => height = 6 units.

Correct Answer: B — 6 units

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Q. The lengths of the sides of triangle ABC are 7 cm, 24 cm, and 25 cm. What type of triangle is it?

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

Solution

Since 7^2 + 24^2 = 25^2, triangle ABC is a right triangle.

Correct Answer: C — Right

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Q. What is the area of an equilateral triangle with side length 'a'?

A. (√3/4)a²
B. (1/2)a²
C. (√2/2)a²
D. (3/2)a²

Solution

The area of an equilateral triangle is given by the formula (√3/4)a².

Correct Answer: A — (√3/4)a²

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Q. What is the circumradius of a triangle with sides 5, 12, and 13?

A. 6.5
B. 7
C. 8
D. 9

Solution

For a right triangle, the circumradius R = hypotenuse/2 = 13/2 = 6.5.

Correct Answer: B — 7

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Q. What is the circumradius of a triangle with sides 6, 8, and 10?

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

Solution

Circumradius R = (abc)/(4K), where K is the area. Area K = 24 (using Heron's formula). R = (6*8*10)/(4*24) = 5.

Correct Answer: A — 5

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Q. What is the circumradius of a triangle with sides 7, 24, and 25?

A. 12.5
B. 13
C. 14
D. 15

Solution

The circumradius R of a triangle can be calculated using the formula R = (abc)/(4 * Area). Here, Area = 84 cm², so R = (7 * 24 * 25)/(4 * 84) = 13.

Correct Answer: B — 13

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Q. What is the circumradius of an equilateral triangle with side length a?

A. a/√3
B. a/2
C. a/√2
D. a/√3

Solution

The circumradius R of an equilateral triangle is given by R = a/(√3).

Correct Answer: D — a/√3

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Showing 31 to 60 of 67 (3 Pages)

Properties of Triangles

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