TS EAMCET · Maths · Properties of Triangles
In a \(\triangle A B C, r_1, r_2\) and \(r_3\) respectively denote the radius of excircles opposite to the vertices \(A\), \(B, C\) and \(r\) denotes the radius of the incircle. If \(p_1, p_2\) and \(p_3\) respectively are the altitudes of the triangle from the vertices \(A, B\) and \(C\), then \(\left(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\right)^2\) is equal to
- A \(\left(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\right)^2 r^2\)
- B \(\frac{1}{r}\left(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\right)\)
- C \(\left(\frac{r}{r_1}+\frac{r}{r_2}+\frac{r}{r_3}\right)^2\)
- D \(r r_1+r r_2+r r_3\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{r}\left(\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}\right)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \Delta=\frac{1}{2} \cdot a \cdot p_1 \\ & \Rightarrow \quad p_1=\frac{2 \Delta}{a} \Rightarrow \frac{1}{p_1}=\frac{a}{2 \Delta} \\ & p_2=\frac{2 \Delta}{b} \Rightarrow \frac{1}{p_2}=\frac{b}{2 \Delta} \\ & p_3=\frac{2 \Delta}{c} \Rightarrow…
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