A beautiful problem on Angle bisector of a triangle and the ration it divides the opposite side.

Hello math bugs(🐞) and hivers(🐝)

Well come to another sweet and interesting geometeic problem. We know that the bisector of angle between two sides of a triangle divides the third and opposite side of the triangle in the ratio of the two sides that conceive the angle. Using this proeprty , we gonna find PQ in the following figure.

Before heading towards solution , we gonna discuss the axioms, postulate or the properties that we need here to solve it.

1️⃣ Pythagoras' theorem. Check details.

2️⃣ An angle bisector and how it divides the third side of a triangle.👇

3️⃣ We also gonna need to solve a simple equation. We will check it in the end.

Solution:

First, we have to cosider ∆ABD to find value of BD. Check the following figure:👇

As we already find BD, CD will be BC - BD = (14 - 5) cm = 9 cm

👉 We must have very familiar to rario and its operation. For example if you are asked to divide $50 between A and B in the ratio 3 : 2, you can easily do it. A will get $30 and B will get $20. We will use the same operation to get AQ, DQ, AP and DP.

Now we can find the value of AQ and DQ. Whicn can be driven in the following way.👇

Again we can find the same way the values of AP and DP. Check it below:👇

👉 We are now at the bottom line. We just need to solve a simple equation to find the value of PQ. Check AD can be written as sum of AP and DP and again it can be written as sum of AQ and DQ. So, we got here a simple equation.

That is given by (AP + DP) = (AQ + DQ). Now check the following figure for the final touch.👇

👉If you have good command over ratio and its operation, the problem can be solved without pen and paper. But Before that we need to know pythagoren triplet as here we needed 5,12 amd 13. So without any calculation we could find the third side of ∆ABD. And then using ratio we could reach at PQ

📢 All the figures used here are made by me using android application. The figure may not be accurate in measurement, just try considering given data only. If there is any silly mistake , please try to ignore it. Though, I tried my best to put it without any misatke.

I hope you liked my today's sweet math problem and its solution.

Thank you so much for bearing me till end.

Have a nice day.

All is well

Regards: @meta007