Given that triangles 𝐴𝐵𝐶 and 𝐸𝐷𝐶 are similar. Determine the value of 𝑥.
Remember the word “similar” in a maths context has a very specific meaning. In similar triangles, corresponding sides are always in the same ratio. Now in our example, we’ve got two similar triangles and they’re right-angle triangle- right triangles. So they have a hypotenuse; they correspond to each other. They have a longer side connected to the right angle and they correspond to each other. And they have a shorter side connected to the right angle and they correspond to each other.
So by saying corresponding sides are always in the same ratio, we can say the ratio, for example, of side 𝐴𝐶 to 𝐸𝐶 is the same ratio as 𝐴𝐵 to 𝐸𝐷. And that’s the same as the ratio from 𝐶𝐵 to 𝐶𝐷. So the question tells us they’re similar triangles, so we know this to be true. So let’s pick some ratios which are actually going to help us solve the problem. So if I take side 𝐸𝐷 and divide that by side 𝐶𝐷, that’s gonna give me the same answer as taking side 𝐴𝐵 and dividing by length 𝐶𝐵.
Now let’s plug the corresponding numbers in. And we can see that 𝑥 divided by thirty is gonna give us the same result as fifty-one divided by thirty-four. Now I can multiply both sides of my equation by thirty to get 𝑥 on its own. And that cancels out the thirty on the left-hand side and gives me fifty-one over thirty-four times thirty on the right-hand side. Now I’m gonna write thirty as a fraction: thirty over one.
And now I can do a bit of cancelling down. Now I can see that thirty-four is divisible by two; so that would be seventeen. And thirty is divisible by two; so that will be fifteen. And then I know that three times seventeen is fifty-one. So seventeen is going to seventeen once and seventeen is going to fifty-one three times. So this becomes 𝑥 is equal to three times fifteen, which is forty-five. So my answer is 𝑥 equals forty-five. Now just one quick thing before we go: if you wrote 𝑥 equals forty-five centimetres, technically you’d be wrong. Look here in the diagram. It tells us 𝑥 is the number of centimetres. So if you said 𝑥 was forty-five centimetres, then we’d be saying that that height there is forty-five centimetres centimetres and that’s not quite right. So 𝑥 equals forty-five.