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Question
Figure shows Δ PQR in which ST || QR and SR and QT intersect each other at M. If `"PT"/"TR" = 5/3` find `("Ar" (triangle "MTS"))/("Ar" (triangle "MQR"))`
Solution
Given : `"PT"/"TR" = 5/3`
To find : `("Ar" (triangle "MTS"))/("Ar" (triangle "MQR"))`
Sol : In Δ PST and Δ PRQ
∠ PST = ∠ PQR
∠ PST = ∠ PQR ...(Corresponding angles)
∴ Δ PST ∼ Δ PQR ....(AA corollary)
∴ `"PT"/"PR" = "ST"/"QR" = 5/8` .....(similar sides of similar triangles)
Now, In Δ MTS and Δ MQR
∠ MTS = ∠ MQR ...(Alternate interior angles)
∠ MTS = ∠ MQR
∴ Δ MTS ∼ Δ MQR ....(AA corollary)
`therefore ("Ar" (triangle "MTS"))/("Ar" (triangle "MQR")) = "TS"^2/"QR"^2 = (5/8)^2 = 25/64`
i.e. 25 : 64
[The ration of areas of two similar triangle is equal to the ratio of square of their corresponding sides.]
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