Q. A circle is inscribed in a triangle. If the sides of the triangle are 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 radius r of the inscribed circle can be found using the formula r = A/s, where A is the area and s is the semi-perimeter. The semi-perimeter s = (7 + 8 + 9)/2 = 12 cm. The area A can be calculated using Heron's formula: A = √[s(s-a)(s-b)(s-c)] = √[12(12-7)(12-8)(12-9)] = √[12*5*4*3] = √720 = 12√5. Thus, r = A/s = (12√5)/12 = √5 cm, which is approximately 2.24 cm.
Q. If a tangent is drawn to a circle from a point outside the circle, what is the relationship between the radius and the tangent at the point of contact?
A.
They are equal
B.
They are perpendicular
C.
They are parallel
D.
They are collinear
Solution
The tangent to a circle at any point is perpendicular to the radius drawn to the point of contact.
Q. If the radius of a circle is doubled, by what factor does the area of the circle increase?
A.
1
B.
2
C.
4
D.
8
Solution
The area of a circle is given by A = πr². If the radius is doubled (r' = 2r), the new area A' = π(2r)² = 4πr². Thus, the area increases by a factor of 4.
Q. In triangle ABC, if angle A = 60 degrees, angle B = 70 degrees, and side a = 10 cm, what is the length of side b using the Law of Sines?
A.
8.66 cm
B.
9.15 cm
C.
7.84 cm
D.
10.00 cm
Solution
Using the Law of Sines: a/sin(A) = b/sin(B). Thus, b = a * (sin(B)/sin(A)) = 10 * (sin(70)/sin(60)). Calculating gives b ≈ 10 * (0.9397/0.8660) ≈ 10.80 cm.
