Question

Prove that the ratio of the areas of two similar triangles are equal to the ratio of the squares of their corresponding sides. Use this theorem to find the ratio of the areas of two similar triangles whose sides are in the ratio 5 : 12.

Answer 1

Answer:

hence proved

Step-by-step explanation:

Given , 2 triangles ABC and PQR  such that ABC similar PQR.

Draw AM perpendicular BC in ABC   and PN perpendicular QR.

We want to prove , ar(ABC)/ar(PQR)=(AB/PQ)²=(BC/QR)²=(AC/PR)²

proof:

        ar (ABC)=1/2×BC×AM

        ar (PQR)=1/2×QR×PN

       ar(ABC)/ar(PQR)=BC/QR=AM/PN -----------------(1)

given ABC similar PQR

         AB/PQ=BC/QR=AC/PR-------------------------------(2)

  also, ∠B=∠Q

           ∠AMB=∠PNQ=90°

           ΔABM similar ΔPQN(AA)

             AB/PQ=AM/PN---------------------------------------(3)

 FROM (1),

          ar(ABC)/ar(PQR)=BC/QR×AM/PN

                                      =AB/PQ×AB/PQ(from (2) and(3))

                                      =(AB/PQ)²

Since AB/PQ=BC/QR=AC/PR

   ar (ABC)/ar(PQR)=(AB/PQ)²=(BC/QR)²=(AC/PR)²

       HENCE THE PROOF.