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प्रश्न
The two similar triangles are equal in area. Prove that the triangles are congruent.
उत्तर
Given: ΔABC ∼ ΔPQR and are equal in area
To prove: ΔABC ≅ ΔPQR
Proof: ∵ ΔABC ∼ ΔPQR
∴ `(Area ΔABC)/(Area ΔPQR) = (AB^2)/(PQ^2) = (BC^2)/(QR^2) = (AC^2)/(PR^2)`
But Area ΔABC = Area ΔPQR ...(Given)
∴ `(AB^2)/(PQ^2) = (BC^2)/(QR^2) = (AC^2)/(PR^2) = 1`
`\implies (AB^2)/(PQ^2) = 1,`
`\implies` AB2 = PQ2
`\implies` AB = PQ
Similarly, BC = QR and AC = PR
Now in ΔABC and ΔPQR
AB = PQ, BC = QR, AC = PR ...(Proved)
∴ ΔABC ≅ ΔPQR ...(SSS criterion of congruency)
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