Question

The areas of two similar triangles are 81 cm² and 49 cm² respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians?

Answer 1

SOLUTION :  

Given : Area of two similar triangles is 81cm² and 49cm² .

(i) Since, the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes.

ar(∆1)/ar(∆2) = (altitude 1/ altitude 2)²

81/49 = (altitude 1/ altitude 2)²

On taking square root on both sides,

√81/49 = √(altitude 1/ altitude 2)²

9/7 = altitude 1/ altitude 2

Altitude 1: altitude 2 = 9 : 7

Hence, the ratio of their corresponding Heights (altitudes) is 9 : 7.

 

(ii) Since, the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding medians.

ar(∆1)/ar(∆2) = (median1/median2)²

81/49 = (median1/median2)²

On taking square root on both sides,

9/7 = median 1/median2

Median 1: median 2 = 9: 7

Hence, the ratio of their corresponding medians is 9 : 7.

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Answer 2

that, when two triangles are similar the ratio of their area is equal to the ratio of square of their corresponding altitudes and corresponding medians.

$\frac{area \: of \: first \: triangle \: a}{area \: of \: second \: triangle} = \: \frac{(area \: of \: first \: }{area \: of \: altitude squaring } \ \\ \\ \frac{81}{49} = \frac{area \: of \: triangle}{area \: of \: altitude} \\ \\ \frac{9}{7} = \frac{altitude \: of \: first \: triangle}{altitude \: of \: second \: triangle} \\ \\ similarly \: we \: get \\ \\ \frac{median \: of \: first \: triangle}{median \: of \: second \: triangle} \: \frac{9}{7}$

$<b><u>$ we did a squaring of altitude of triangle on LHS see above means ( altitude of triangle) ²

then we did both sides squaring after squaring on both sides 81 /49 means √81 /49 = 9/7 come
Altitude of triangle = 9/7

then the corresponding height of the ratio is

9:7

Corresponding medians is also 9:7

Then in corresponding of medians we do same thing

in this we square the (Corresponding of median of triangle)²

in this same we do 81 /49 square on both sides

√81 /49 = median of triangle
= 9/7

so, the corresponding medians in the ratio is 9:7

Hope it helps!