Question

Prove that "when two triangles are similar , the ratio areas of those triangle is equal to the ratio of the squares of their corresponding sides".

Plz give full answer. ​

Answer 1

see the attached photo.

$\longrightarrow$ As We Know To Find The Area We Need The Height Of The Triangle. So We Can Draw A Perpendicular AD From A To BC And PS From P To QR.

$\longrightarrow$ In Triangle ABD And Triangle PQS

• ∠B =∠Q {∵ΔABC∼ΔPQR}

• ∠ADB =∠PSQ=90∘ (By Construction)

$\longrightarrow$ As Two Angles Are Equal So The Third Angle Of Both Triangles Should Also Be Equal.

• ∠BAD=∠QPS

$\longrightarrow$ So By AAA Similarity

• ΔABD∼ΔPQS

$\longrightarrow$ So We Can Say The Ratio Of Corresponding Sides Should Be Equal. So We Can Write,

• $\sf \frac{AB}{PQ}=\frac{AD}{PS}………………..(i)$

$\longrightarrow$ Now We Can Write Area Of Triangle ABC as

• $\sf Area(ΔABC)=\frac{1}{2}×AD×BC ………………..(ii)$

• $Area(ΔPQR)=\frac{1}{2}×PS×QR………………..(iii)$

$\longrightarrow$ On Dividing Equation (ii) and (iii)

• $\sf \frac{Area(ΔABC)}{Area(ΔPQR)}=\frac{AD×BC}{PS×QR}$

$\longrightarrow$ By Using Equation (i) We Can Write

• $\sf \frac{Area(ΔABC)}{Area(ΔPQR)}= \frac{AB×BC}{PQ×QR} ………………..(iv)$

$\longrightarrow$ As given ΔABC∼ΔPQR

$\longrightarrow$ So This Ratio Of Corresponding Sides Should Be Equal. So We Can Write,

• $\sf \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}$

$\longrightarrow$ Hence We Can Use This Value In Equation (Iv). So We Can Write

• $\sf \frac{Area(ΔABC)}{Area(ΔPQR)} = \frac{AB×AB}{PQ×PQ} = \bigg(\frac{AB}{PQ}\bigg)²$

$\longrightarrow$ Similarly We Can Write

• $\sf \frac{Area(ΔABC)}{Area(ΔPQR)} = \bigg(\frac{AB}{PQ}\bigg)² = \bigg(\frac{BC}{QR}\bigg)²= \bigg(\frac{AC}{PR}\bigg)²$

$\longrightarrow$ Hence We Can Say The Ratio Of The Areas Of Two Similar Triangles Is Equal To The Ratio Of The Square Of Their Corresponding Sides.

• NOTE: In General Area(a) Of Any Triangle Is

• $\sf A= \frac{1}{2}×base×height$

$\longrightarrow$ That’s Why We Need To Construct Perpendicular Triangles For Height.

I hope it helps you ❤️✔️