Question

In the given figure $AB = 9$ , $BC = 12$ and $AC = 15$ . $BP \bot AC$ . Find $BP$ and $AP$ respectively.

A) $8.4,4.3$
B) $7.2,5.4$
C) $6.9,3.8$
D) $5.5,5.8$

Answer 1

Hint: We have two different right-angled triangles in the given figure so we will use the properties of the right-angled triangles to solve the problem. We will use two different relations and then the given values to simplify the problem so that the answer is obtained.

Complete step by step solution:
First of all, understand the given data correctly.
We have already given the length of all sides of the triangle.
The given lengths are \[AB = 9\] , $BC = 12$ and $AC = 15$ .
Also, additionally it is given that the segment $BP$ makes a right angle on the side $AC$ .
Therefore, using the property of the right-angled triangle we can write the following in case of the right-angled triangle.
$B{P^2} = AP \times PC$...................… (1)
Similarly, we can write the other relation as follows:
$A{B^2} = AP \times AC$.............… (2)
We have already given the length of $AB$ and $AC$.
Substituting these values in equation (2) we can write:
$\Rightarrow {9^2} = AP \times 15$
Simplifying the above equation for the unknown side we get,
$\Rightarrow AP = 5.4$
Now we will use this value to obtain the other obtained value.
Now that from the diagram it can be observed that,
$\Rightarrow AC = AP + PC$
Now we can write the above relation as:
$\Rightarrow 15 = 5.4 + PC$
Hence,
$\Rightarrow PC = 9.6$
Now we can make a use of the equation number (1) as follows:
$\Rightarrow B{P^2} = 5.4 \times 9.6$
Therefore, on multiplying we get,
$\Rightarrow B{P^2} = 51.84$
Now as it is the length of a side, we will take the positive square root.
$\Rightarrow BP = 7.2$
Therefore, we have now obtained the length of all the required sides and we get $AP = 5.4$ and $BP = 7.2$.

Hence the correct option is C.

Note: We only used two basic relations which are obtained by making keen observations about a right-angled triangle. Later part mostly contains the calculations but to use the relations in a proper way is the most important step. We also use the given data by making the adjustments that were necessary.