Relation between Areas and Sides of Similar Triangles
"The ratio of...
Question
"The ratio of areas of similar triangles is equal to the ratio of squares of corresponding side". Prove by using equilateral triangle scelane triangles.
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Solution
Ratio of area of similar △ is = ratio of square of corresponding sides
△ABCsin△PQR
ar(ABC)=12×B×H=12×BC×AK ----- (1)
ar(PQR)=12×B×H=12×QR×PL ----- (2)
Dividing (1) and (2)
ar(ABC)ar(△PQR)=BC×AMQR×PN ------ (3)
In △ABK and △PQL
∠B=∠Q (angle of similar △ are equal )
∠M=∠N ( Both 90o)
△ABKsin△PQL ( By AA similarity )
∴ABPQ=AMPN {By CPCT } ------ (4)
⇒ar(△ABC)ar(△PQR)=BCQR×ABPQ ------- (5) {from (3) and (4) }