Triangle ABC is right angle at A. The points P and Q are on hypotenuse BC such that `BP = PQ = QC`.if `AP = 3 and AQ = 4`, then length BC is equal to

Options

A. `3sqrt(5)`
B. `5sqrt(3)`
C. `4sqrt(5)`
D. 7

Answer

A

Explanation

Let BP = PQ = QC = x
In `Delta ABP`, using cosine rule
`9=c^(2)+x^(2)-2cx cos B`
But `cos B = (c )/(3x)`
`rArr 9 = x^(2)+(c^(2))/(3)` ….(1)
Similarly using rule in `Delta ACQ`, we get
`16 = x^(2)+(b^(2))/(3)` .......(2)
Adding (1) and (2), we get `25=2x^(2)+(b^(2)+c^(2))/(3)`
`therefore 25=2x^(2)+((3x)^(2))/(3)`
`therefore 25=2x^(2)+3x^(2)`
`therefore x^(2)=5`
`therefore BC = 3x = 3sqrt(5)`