In a triangle, the two longest sides are 13 cm and 12 cm. The angles of the triangle are in arithmetic progression. The radius of the circle inscribed in this triangle is:
Explanation:
Let the three angles be a − d, a and a + d. ∴ (a − d) + a + (a + d) = 180 ∴ a = 60.
Since (a + d) is the largest angle it will be opposite the largest side i.e., 13 and a will be opposite second largest side i.e., 12.
The triangle along with the angles are shown below.
Applying the cosine rule, we get; Cos 60 = (x2 + 132 − 122)/(2 × x × 13) ∴ x = 10.65, 2.34. Inradius = Area/Semiperimeter. When x = 10.65, the semiperimeter = (10.65 + 13 + 12)/2 = 17.825. ∴ Inradius = [(1/2) × 10.65 × 13 × Sin60]/17.825 = 3.36. When x = 2.34, the semiperimeter = (2.34 + 13 + 12)/2 = 13.67. ∴ Inradius = [(1/2) × 2.34 × 13 × Sin60]/13.67 = 0.963 ≈ 1. Hence option (d).
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