First slide

Properties of triangles

Question

In $Δ A B C \cos A + \cos B + \cos C = \frac{7}{4}$ and $\frac{r}{R} = \frac{k}{4}$ , then $k =$

Moderate

A

1
B

2
C

3
D

4

Solution

$\begin{array}{l} \cos A + \cos B + \cos C = \frac{7}{4} \\ \Rightarrow 1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} = \frac{7}{4} (i f A + B + C = π, then cosA + cosB + cosC = 1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}) \\ \Rightarrow 4 \sin \frac{A}{2} \sin \frac{C}{2} \sin \frac{B}{2} = \frac{3}{4} \\ \Rightarrow \frac{r}{R} = \frac{3}{4} (∵ 4 R \sin \frac{A}{2} \sin \frac{C}{2} \sin \frac{B}{2} = r) \\ \Rightarrow k = 3 \end{array}$

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