If the areas of two similar triangles are equal then prove that they are congruent
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Two triangles ABC and PQR are similar
Then by area of similar triangles theorem;
ar(ABC)/ar(PQR) = (AB/PQ)^2 = AB^2/ PQ^2
It is given that area of ABC and PQR are equal
So,
ar(ABC)/ar(PQR) = 1
Which means,
AB^2 = PQ^2
AB = PQ
This is true for other two sides as well so;
AB = PQ
BC = QR
AC = PR
By sss congruency,
ABC is congruent to PQR
Then by area of similar triangles theorem;
ar(ABC)/ar(PQR) = (AB/PQ)^2 = AB^2/ PQ^2
It is given that area of ABC and PQR are equal
So,
ar(ABC)/ar(PQR) = 1
Which means,
AB^2 = PQ^2
AB = PQ
This is true for other two sides as well so;
AB = PQ
BC = QR
AC = PR
By sss congruency,
ABC is congruent to PQR
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