Q. A circle is inscribed in a triangle. If the triangle has sides of lengths 7 cm, 8 cm, and 9 cm, what is the radius of the inscribed circle?
A.
3 cm
B.
4 cm
C.
5 cm
D.
6 cm
Solution
The area A of the triangle can be calculated using Heron's formula. The semi-perimeter s = (7 + 8 + 9) / 2 = 12. The area A = √(s(s-a)(s-b)(s-c)) = √(12(12-7)(12-8)(12-9)) = √(12*5*4*3) = 12√5. The radius r = A/s = (12√5)/12 = √5 cm, which is approximately 4 cm.
Q. A circle is inscribed in a triangle. What is the radius of the incircle if the triangle has sides of lengths 7, 8, and 9 units?
A.
4 square units
B.
3 square units
C.
5 square units
D.
2 square units
Solution
The area of the triangle can be calculated using Heron's formula. The semi-perimeter s = (7+8+9)/2 = 12. The area A = √(s(s-a)(s-b)(s-c)) = √(12(12-7)(12-8)(12-9)) = √(12*5*4*3) = 12√5. The radius r = A/s = 12√5/12 = √5. The radius is approximately 3 square units.
Q. If the coordinates of the vertices of a triangle are (0, 0), (4, 0), and (0, 3), what is the area of the triangle?
A.
6 square units
B.
12 square units
C.
8 square units
D.
10 square units
Solution
The area of a triangle can be calculated using the formula A = 1/2 * base * height. Here, base = 4 and height = 3, so A = 1/2 * 4 * 3 = 6 square units.
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.
