Question

What is your favorite movie name? (Bollywood, Hollywood) And Prove that the ratio of the area of two similar triangles is equal to the ratio of the squares of their corresponding angle -bisector segments.

Answer 1

Answer

It is proved ≈

The ratio of the areas of two similar triangles is equal to the ratio of the square of their corresponding sides.

( my favorite actor is akshay kumar)

Answer 2

$\huge\mathtt\red{Question}$

Prove that the ratio of the area of two similar triangles is equal to the ratio of the squares of their corresponding angle -bisector segments.

$\huge\mathtt\red{Solution}$

$\mathtt\pink{We\:know\:that,}$

Ratio of area of similar △ is = ratio of square of corresponding side.

△ABC sin △PQR

$ar(ABC) = \frac{1}{2} \times b \times h \\ = \frac{1}{2} \times bc \times ak................(1) \\ ar(PQR) = \frac{1}{2} \times b \times h \\ = \frac{1}{2} \times qr \times pl..................(2)$

Dividing (1) and (2) , we get,

$\frac{ar(ABC)}{ar(△PQR)} = \frac{bc \times am}{qr \times pn}................(3)$

In △ABK and △PQL,

$\longrightarrow$ ∠B = ∠Q (angles of similar △a are equal)

$\longrightarrow$ ∠M = ∠N ( Both 90°)

△ABK sin △PQR

$\frac{ab}{pq} = \frac{ab}{qr} = \frac{ac}{pr} \\ → \frac{ar(△ABC)}{ar(△PQR)} = \frac{AB \times AB}{PQ \times PQ} = ( \frac{AB}{PQ} )^{2} \\ →( \frac{BC}{QR} )^{2} = ( \frac{AC}{PR} )^{2}$

Hence proved.