1.

A triangle has sides 6,7, and 8. The linethrough its incenter parallel to the shortest side is drawn to meet the othertwo sides at P and Q. Then find the length of the segment PQ.

Answer» We can draw the diagram with the given details.
Please refer to video to see the diagram.
Now, `Delta = r**s = r**(a+b+c)/2 = r**(6+7+8)/2 = 21/2r`
`:. Delta = 21/2 r->(1)`
From the diagram, we can see that,
`Delta = 1/2**h**BC = 1/2**h**6 = 3h`
`:. Delta = 3h->(2)`
From (1) and (2),
`=>21/2r = 3h => r/h = 2/7`
Now, `Delta APQ and Delta ABC` are similar triangles.
`:. (h-r)/h = (PQ)/(BC)`
`=>1-r/h = (PQ)/6`
`=>6(1-2/7) = PQ`
`=>30/7 = PQ`
Therefore, length of `PQ` is `30/7.`


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