Both 𝐴𝐵𝐷 and 𝐷𝐵𝐶 are right-angled triangles. Work out the value of 𝑥. Give your answer correct to two decimal places.
The clue here is given in the first sentence. We’re told these are right-angled triangles. That means we can either use Pythagoras’s theorem or trigonometry to help us find the
missing dimensions. In fact, aside from the right angles, we know no other angles in these triangles nor
do we need to find them, which tells us that trigonometry is not required.
Pythagoras’s theorem says that the square of the longest side — that’s the hypotenuse
— is equal to the sum of the squares of the two shorter sides. That’s usually written as 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐
is the hypotenuse or the longest side. We can find the longest side by looking for the side that’s directly opposite the
Let’s first consider triangle 𝐷𝐵𝐶. We are given the lengths of the two shorter sides in this triangle and we want to
find the length of the side 𝐷𝐵. That’s the hypotenuse, which we can call 𝑦. We can substitute these values into our formula for Pythagoras’s theorem to get 12
squared plus 16 squared is equal to 𝑦 squared. Evaluating 12 squared and 16 squared gives us 144 and 256. 144 plus 256 is 400.
To solve this equation, we want to do the opposite of squaring — that’s finding the
square root of both sides of the equation. The square root of 400 is 𝑦. So 𝑦 is equal to 20. And that means we can change the value of 𝑦 in our diagram, giving us that 𝐷𝐵 is
20 millimeters long.
Now, let’s consider the second triangle in our diagram — that’s triangle 𝐴𝐵𝐷. We’ve calculated a second side in this right-angled triangle. Once again, we can use Pythagoras’s theorem to find the length 𝐴𝐵, which is
labelled as 𝑥 millimeters. Since 𝐴𝐵 is the hypotenuse, substituting our values into the formula for
Pythagoras’s theorem gives seven squared plus 20 squared is equal to 𝑥 squared.
Seven squared is 49 and 20 squared is 400, which means that 𝑥 squared is equal to
449. Once again, we’ll find the square root of both sides of the equation, which gives us
𝑥 is equal to the square root of 449. Popping that into our calculator, we get that 𝑥 is equal to 21.189. Correct to two decimal places, 𝑥 is equal to 21.19.
Remember whilst we usually try to include units with our answer, for this one, we
were asked to find the value of 𝑥. And 𝑥 was defined as the number of millimeters. If we’d written that 𝑥 was equal to 21.19 millimeters, then technically we’d be
saying that the length is 21.19 millimeters millimeters, which doesn’t really make
an awful lot of sense. In this case then, we leave our answer just as 𝑥 is equal to 21.19.