Question

explain the theorem of areas of similarity Triangle​

Answer 1

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$\bold{\underline{\underline{QUESTION}}}$

EXPLAIN THE THEOREM OF AREAS OF SIMILARITY TRIANGLE.

$\bold{\underline{\underline{ANSWER}}}$

Theorem: If two triangles are similar, then the ratio of the area of bothtriangles is proportional to the square of the ratio of their corresponding sides. This proves that the ratio ofareas of two similar triangles is proportional to the squares of the corresponding sides of both the triangles.

$\bold{\underline{\underline{Step\:To\:Step\:Explanation}}}$

Before knowing areas of a similar triangle, the two triangles are similar  to each other if,

i)    Corresponding angles of the triangles are equal,

ii)   Corresponding sides of  the triangles are in  proportion

Thus, two triangles ΔABC and ΔPQR are similar if,

i)    ∠A=∠P, ∠B=∠Q and ∠C=∠R

ii)    ABPQ = BCQR  = ACPR

If we have two similar triangles, then not only their angles and sides share a relationship but also the ratio of their perimeter, altitudes, angle bisectors, areas and other aspects are in ratio.

In the upcoming discussion, the relation between the areas of two similar triangles is discussed.

Theorems on the area of similar triangles

Theorem: If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides.

To prove this theorem, consider two similar triangles ΔABC and ΔPQR;

According to the stated theorem,

area of ΔABCarea of ΔPQR = (ABPQ)2 =(BCQR)2 = (CARP)2

As, Area of triangle = 12 × Base × Height

To find the area of ΔABC and ΔPQR, draw the altitudes AD and PE from the vertex A and P of ΔABC andΔPQR as shown in the figure given below:

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