Question Video: Applying the Pythagorean Theorem to Solve More Complex Problems Mathematics • First Year of Preparatory School

Video Transcript

Find the perimeter of 𝐴𝐵𝐶𝐷.

We’ve been asked to find the perimeter of this quadrilateral, which is the sum of all its side lengths. That’s 𝐴𝐵 plus 𝐵𝐶 plus 𝐶𝐷 plus 𝐷𝐴. We’ve been given the lengths of three of these sides, but one side length, 𝐷𝐴, is unknown. We’ll need to use the information we’ve been given to calculate this length.

Looking more closely at the figure, we can see that the quadrilateral 𝐴𝐵𝐶𝐷 has been divided into two right triangles: triangle 𝐴𝐵𝐶 and triangle 𝐴𝐶𝐷. We can calculate the length of any side in a right triangle if we know the lengths of the other two sides by applying the Pythagorean theorem. This states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. If we label the two shorter sides as 𝑎 and 𝑏 and the hypotenuse as 𝑐, then this can be expressed as 𝑎 squared plus 𝑏 squared equals 𝑐 squared. But in the pink triangle, in which side 𝐷𝐴 is the hypotenuse, we’re only given the length of one of the other sides. Side length 𝐴𝐶 is also unknown.

However, this side is common with triangle 𝐴𝐵𝐶 in which we do know both of the other two side lengths. So, our approach will be to first calculate the length of 𝐴𝐶 using the Pythagorean theorem in triangle 𝐴𝐵𝐶 and then use this to apply the Pythagorean theorem a second time in triangle 𝐴𝐶𝐷. 𝐴𝐶 is the hypotenuse of triangle 𝐴𝐵𝐶, so applying the Pythagorean theorem gives the equation 𝐴𝐶 squared equals 20 squared plus 48 squared. Evaluating each of the squares and finding their sum gives 𝐴𝐶 squared equals 2704.

Now, we could continue and find the length of 𝐴𝐶 by square rooting, but actually we don’t need to. When we apply the Pythagorean theorem in triangle 𝐴𝐶𝐷, in which 𝐷𝐴 is the hypotenuse, we obtain the equation 𝐷𝐴 squared equals 𝐶𝐷 squared plus 𝐴𝐶 squared. We know 𝐶𝐷 is 39 centimeters, and we’ve already found that the squared value of 𝐴𝐶 is 2704. So rather than square rooting to find 𝐴𝐶 and then squaring again in our second application of the Pythagorean theorem, we can just substitute the value of 𝐴𝐶 squared directly. That gives 𝐷𝐴 squared equals 39 squared plus 2704. Evaluating 39 squared gives 1521, and then adding 2704 gives 4225. Taking the square root of both sides of this equation gives 𝐷𝐴 equals 65. We’re only interested in the positive solution here as 𝐷𝐴 is a length.

So, having calculated the length of the final side in quadrilateral 𝐴𝐵𝐶𝐷, we’re now ready to find its perimeter. Substituting the value of 65 into our sum for the perimeter gives 20 plus 48 plus 39 plus 65, which is 172. By applying the Pythagorean theorem twice, we’ve found the length of the final side of this quadrilateral and hence its perimeter, which is 172 centimeters.