# Question Video: Using the Congruency between Two Triangles to Solve Linear Equations and Find a Side Length Mathematics • 8th Grade

Find ππ in the given figure.

04:47

### Video Transcript

Find ππ in the given figure.

Now looking at the diagram, we can see that we have two triangles, triangles πππ and triangles πππ. ππ is the length of the line joining points π and π. And we can see that we donβt know the length of this line, but weβve been given it in terms of a variable π₯ whose value we donβt yet know. In order to answer this question, weβre going to need to find a way to work out the value of π₯ so that we can then calculate the length of ππ.

Weβre going to approach this question using congruent triangles. And what we want to show, to start off with, is that triangles πππ and πππ are congruent to each other. Letβs have a look at them more closely. Now theyβre both right-angled triangles. We can see that because the right angle has been marked on one of the two triangles, and the line ππ is a straight line, which means the angle on the other side is also a right angle. So when weβre thinking about proving congruence of these two triangles, weβll use the congruence theorems for right-angled triangles.

So first, letβs see what we know about these two triangles. Well, the hypotenuses of the two triangles are the same. We can see this because of the blue lines that have been used to indicate two lines are the same length. So we have the statement that side ππ is equal in length to side ππ. We use the letter H to indicate that these are the hypotenuses of the two triangles.

Now letβs look at another side in the two triangles, the side ππ. ππ is a shared side as it is in both triangles. So this means that a second side of the two triangles is equal in length. Weβll refer to this as L, a leg of the triangles. Now we recall our congruence theorems for right-angled triangles, one of which is this: if the hypotenuse and leg of a right-angled triangle are congruent to the hypotenuse and corresponding leg of another right-angled triangle, then the two triangles are congruent. Therefore, weβve shown that triangle πππ is congruent to triangle πππ, and the notation HL shows us that itβs the hypotenuse leg theorem of a right-angled triangle that weβre using.

What all of this means then is that if these two triangles are congruent to each other, then the third sides must be equal in length, i.e., side ππ is equal to side ππ. Each of these sides unknown in terms of the variable π₯, so this means we can set up an equation involving π₯. Our equation is that four π₯ minus one is equal to three π₯ plus two. Weβre now able to solve this equation in order to find the value of π₯.

The first step is to subtract three π₯ from each side of the equation. Doing so, gives π₯ minus one is equal to two. Next, I need to add one to both sides of the equation. And in doing so, I now have that π₯ is equal to three. So we found the value of π₯. We havenβt finished the question, however. The question asked us to find the length of ππ.

So letβs return to the diagram and letβs look at an expression for ππ. ππ can be found by adding ππ to ππ. In terms of π₯, this means weβre adding three π₯ plus two to four π₯ minus one. A simplified expression for ππ is therefore seven π₯ plus one. Now to find the length of ππ, we need to substitute the value of π₯ that weβve just calculated, π₯ is equal to three. So we have ππ is equal to seven multiplied by three, add one, and this gives us our answer of 22 or 22 units for the length of ππ.

So within this question, we didnβt dive straight in to forming and solving the equation. We had to first prove that the two triangles are congruent to each other by remembering the different congruence theorems for right-angled triangles. Only once weβd done that, were we able to setup and solve the equation in order to work out the length of ππ.
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