Question

Please solve q 85 ?

Answer 1

9

Explanation:

For the triangles `Delta PAC` and `Delta PAB` to be similar, all three angles must be the same.

• It is easy to see that `/_BPA` is common
• `/_ACP` is obtuse - so it can not be equal to `/_ABP`, and thus it must be equal to the remaining angle, `/_PAB`
• Finally, we must have `/_PAC=/_ABP`

By the property of similar triangles we have

`AC : AP:PC = AB:BP:AP`

Denoting `PC` and `AP` by `x` and `y`, respectively we have (using `BP = BC+PC = 7+x`)

`6/8 = y/(7+x) = x/y`

Hence `y^2 = x(7+x)` and we have

`6^2/8^2 = x^2/y^2 = x/(7+x)`

and thus

`x/(7+x) = 9/16 qquad implies qquad x/7 = 9/(16-9) = 9/7`

Thus
`x = 9`