### Video Transcript

Find ππ in the given figure.

Now looking at the diagram, we can
see that we have two triangles, triangles π
ππ and triangles π
ππ. 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 π
ππ and π
ππ 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
π
π is equal in length to side π
π. 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 π
π. π
π 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 π
ππ is congruent to triangle π
ππ, 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.