Properties of Triangles
Q. If the angles of triangle ABC 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
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Solution
Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 degrees. Thus, 9x = 180 degrees, x = 20 degrees. The largest angle is 4x = 80 degrees.
Correct Answer: B — 80 degrees
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Q. If the angles of triangle DEF are in the ratio 2:3:4, what is the measure of the largest angle?
A.
40 degrees
B.
60 degrees
C.
80 degrees
D.
90 degrees
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Solution
Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180. So, 9x = 180, x = 20. The largest angle = 4x = 80 degrees.
Correct Answer: C — 80 degrees
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Q. If the area of triangle ABC is 30 cm² and the base BC = 10 cm, what is the height from A to BC?
A.
5 cm
B.
6 cm
C.
7 cm
D.
8 cm
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Solution
Area = 1/2 * base * height. Therefore, 30 = 1/2 * 10 * height, height = 6 cm.
Correct Answer: B — 6 cm
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Q. If the area of triangle ABC is 30 square units and the base BC = 10 units, what is the height from A to BC?
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Solution
Area = 1/2 * base * height => 30 = 1/2 * 10 * height => height = 30 / 5 = 6.
Correct Answer: A — 5
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Q. If the area of triangle ABC is 30 square units and the base BC is 10 units, what is the height from A to BC?
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Solution
Area = 1/2 * base * height. Thus, 30 = 1/2 * 10 * height. Height = 6.
Correct Answer: B — 5
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Q. If the area of triangle ABC is 60 cm² and the base BC = 12 cm, what is the height from A to BC?
A.
5 cm
B.
10 cm
C.
12 cm
D.
15 cm
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Solution
Area = (1/2) * base * height. Therefore, 60 = (1/2) * 12 * height, height = 10 cm.
Correct Answer: B — 10 cm
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Q. If the area of triangle JKL is 30 cm² and the base JK is 10 cm, what is the height from point L?
A.
3 cm
B.
6 cm
C.
5 cm
D.
4 cm
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Solution
Area = 1/2 * base * height. 30 = 1/2 * 10 * height. Therefore, height = 6 cm.
Correct Answer: C — 5 cm
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Q. If the circumradius of triangle ABC is 10 cm and the area is 48 cm², what is the length of the side opposite to angle A?
A.
12 cm
B.
14 cm
C.
16 cm
D.
18 cm
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Solution
Using the formula R = (abc)/(4 * Area), we can find the side opposite to angle A. Let a = side opposite to A. Then, a = (4 * Area * R) / (bc) = (4 * 48 * 10) / (b * c).
Correct Answer: B — 14 cm
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Q. If the circumradius R of triangle ABC is 5 cm, what is the maximum area of the triangle?
A.
12.5 cm²
B.
15 cm²
C.
20 cm²
D.
25 cm²
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Solution
The maximum area of a triangle with circumradius R is given by the formula Area = (abc)/(4R). For maximum area, the triangle should be equilateral, thus Area = (3√3/4) * (R^2) = (3√3/4) * (5^2) = 25√3/4 cm².
Correct Answer: C — 20 cm²
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Q. If the height of an isosceles triangle is 12 cm and the base is 10 cm, what is the area of the triangle?
A.
60 cm²
B.
70 cm²
C.
80 cm²
D.
90 cm²
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Solution
The area of a triangle is given by (1/2) * base * height = (1/2) * 10 * 12 = 60 cm².
Correct Answer: A — 60 cm²
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Q. If the medians of a triangle are 6 cm, 8 cm, and 10 cm, what is the area of the triangle?
A.
48 cm²
B.
60 cm²
C.
72 cm²
D.
80 cm²
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Solution
Area = (4/3) * √[m1 * m2 * m3] = (4/3) * √[6 * 8 * 10] = 48 cm².
Correct Answer: B — 60 cm²
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Q. If the medians of a triangle are 6, 8, and 10, what is the area of the triangle?
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Solution
Area = (4/3) * √(s(s - m1)(s - m2)(s - m3)), where s = (6 + 8 + 10)/2 = 12. Area = (4/3) * √(12 * 6 * 4 * 2) = 48.
Correct Answer: C — 48
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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
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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
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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
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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
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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°
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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
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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?
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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
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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
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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
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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
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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°
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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
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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
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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
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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°
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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
Show solution
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 area of triangle ABC?
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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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