Video: Using the Congruency between Two Triangles to Solve Linear Equations and Find a Side Length

Find π‘†π‘Š in the given figure.

04:47

Video Transcript

Find π‘†π‘Š in the given figure.

Now looking at the diagram, we can see that we have two triangles, triangles π‘…π‘‡π‘Š and triangles 𝑅𝑇𝑆. π‘†π‘Š is the length of the line joining points 𝑆 and π‘Š. And we can see that we don’t know the length of this line, but we’ve been given it in terms of a variable π‘₯ whose value we don’t yet know. In order to answer this question, we’re going to need to find a way to work out the value of π‘₯ so that we can then calculate the length of π‘†π‘Š.

We’re going to approach this question using congruent triangles. And what we want to show, to start off with, is that triangles π‘…π‘‡π‘Š and 𝑅𝑇𝑆 are congruent to each other. Let’s have a look at them more closely. Now they’re both right-angled triangles. We can see that because the right angle has been marked on one of the two triangles, and the line π‘†π‘Š is a straight line, which means the angle on the other side is also a right angle. So when we’re thinking about proving congruence of these two triangles, we’ll use the congruence theorems for right-angled triangles.

So first, let’s 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‘Š is equal in length to side 𝑅𝑆. We use the letter H to indicate that these are the hypotenuses of the two triangles.

Now let’s look at another side in the two triangles, the side 𝑅𝑇. 𝑅𝑇 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. We’ll 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, we’ve shown that triangle π‘…π‘‡π‘Š is congruent to triangle 𝑅𝑇𝑆, and the notation HL shows us that it’s the hypotenuse leg theorem of a right-angled triangle that we’re 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 π‘‡π‘Š is equal to side 𝑇𝑆. 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. We’re 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 haven’t finished the question, however. The question asked us to find the length of π‘†π‘Š.

So let’s return to the diagram and let’s look at an expression for π‘†π‘Š. π‘†π‘Š can be found by adding 𝑆𝑇 to π‘‡π‘Š. In terms of π‘₯, this means we’re adding three π‘₯ plus two to four π‘₯ minus one. A simplified expression for π‘†π‘Š is therefore seven π‘₯ plus one. Now to find the length of π‘†π‘Š, we need to substitute the value of π‘₯ that we’ve just calculated, π‘₯ is equal to three. So we have π‘†π‘Š is equal to seven multiplied by three, add one, and this gives us our answer of 22 or 22 units for the length of π‘†π‘Š.

So within this question, we didn’t 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 we’d done that, were we able to setup and solve the equation in order to work out the length of π‘†π‘Š.

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