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Question: 42. In a $\triangle ABC$, if $a=10$, $\tan A = \frac{3}{4}$, then $R=$...

  1. In a △ABC\triangle ABC, if a=10a=10, tan⁡A=34\tan A = \frac{3}{4}, then R=R=
A

152\frac{15}{2}

B

252\frac{25}{2}

C

253\frac{25}{3}

D

154\frac{15}{4}

Answer

253\frac{25}{3}

Explanation

Solution

Given: In ΔABC, side a=10a = 10 and tan⁡A=34\tan A = \frac{3}{4}.

  • From tan⁡A=34\tan A = \frac{3}{4}, construct a right triangle with opposite = 3 and adjacent = 4, so the hypotenuse = 5.
  • Thus, sin⁡A=35\sin A = \frac{3}{5}.
  • Using the extended sine law: a=2Rsin⁡A⇒R=a2sin⁡A=102×35=1065=10×56=506=253a = 2R \sin A \quad \Rightarrow \quad R = \frac{a}{2\sin A} = \frac{10}{2 \times \frac{3}{5}} = \frac{10}{\frac{6}{5}} = \frac{10 \times 5}{6} = \frac{50}{6} = \frac{25}{3}