Triangles having equal areas and having one side of one of the triangles, equal to one side of the other, have their corresponding altitudes equal. GIVEN : Two triangles `A B C` and `P Q R` such that: `a r( A B C)=A R( P Q R)` (ii)`AB=P Q`

Explanation

Given : Two triangles `ABC` and `PQR` such that (i) `ar(∆ABC)=ar(∆PQR)` and (ii) `AB=PQ`.
`CN` and `RT` and the altitude corresponding to `AB` and `PQ` respectively of the two triangles.
To prove: `CR = RT`
Proof: In `∆ABC`,
`CN` is the altitude corresponding to the side AB.
`ar(∆ABC)=1/2ABxxCN` ....(i)
Similarly, `ar(∆PQR)=1/2PQxxRT` ....(ii)
From equation (i) and (ii), we get:
`ar(∆ABC)=ar(∆PQR)` [Given]
`∴1/2ABxxCN=1/2PQxxRT`
Also, `AB=PQ` [Given]
`CN=RT`
Hence proved.