# Video: Applications of the Pythagorean Theorem

In this video, we will learn how to apply the Pythagorean theorem in geometric questions and real-life situations.

14:08

### Video Transcript

In this video, we will learn how to apply the Pythagorean theorem in geometric questions and real-life situations. We will begin by recalling what the Pythagorean theorem states.

The Pythagorean theorem 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 the hypotenuse is labeled 𝑐 and the two shorter sides are labeled 𝑎 and 𝑏, then the Pythagorean theorem states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. We will now use this theorem to solve some real-life problems in context.

A man on the top of a building wants to have a guy wire extended to a point on the ground 20 foot from the base of the building. To the nearest foot, how long does the wire have to be if the building is 50 foot tall?

Let’s begin by drawing a diagram. We are told that the building is 50 foot tall. The wire is extended to a point on the ground 20 foot from the base of the building. We need to calculate the length of this wire, which we will call 𝑥. We can see from the diagram that these values create a right triangle. To calculate the missing length in any right triangle, we can use the Pythagorean theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or hypotenuse. The longest side is always opposite the right angle.

Substituting in our values gives us 20 squared plus 50 squared is equal to 𝑥 squared. The values 20 and 50 could be written in either order. 20 squared is equal to 400, and 50 squared is equal to 2500. Adding these gives us a value of 𝑥 squared equal to 2900. Our final step is to square root both sides of this equation. 𝑥 is equal to the square root of 2900. Typing this into the calculator gives us 53.851648 and so on.

We are asked to give our answer to the nearest foot. So, we need to round to the nearest whole number. As the number in the tens column is greater than or equal to five, we round up. The length of the guy wire is 54 foot, to the nearest foot. This is a sensible answer as it is greater than 50 but less than 70. The hypotenuse must be greater than the other two lengths but smaller than the sum of the shorter two lengths.

We will now look at a second question that looks at solving a real-life problem.

The figure shows a 129-meter long bridge on supports 𝑀𝐶 and 𝑀𝐷 attached at the midpoint 𝑀. If 𝐴𝐶 is equal to 51.6 meters, find the length of 𝑀𝐶 to the nearest hundredth.

We are told in the question that the length of the bridge 𝐴𝐵 is 129 meters. As 𝑀 is the midpoint of 𝐴𝐵, we can calculate the distance 𝐴𝑀 by dividing 129 by two. This is equal to 64.5 meters. We are also told that the length 𝐴𝐶 is 51.6 meters. 𝐴𝑀𝐶 is a right triangle where we know two lengths and need to calculate the length 𝑀𝐶.

We can do this using the Pythagorean theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared. 𝑐 is the longest side of the right triangle, known as the hypotenuse. In this case, this is the length 𝑀𝐶. Substituting in our values gives us 64.5 squared plus 51.6 squared is equal to 𝑥 squared. Typing the left-hand side into our calculator gives us 6822.81. We can then square root both sides of this equation to calculate the value of 𝑥. 𝑥 is equal to 62.600302 [82.600302] and on so.

We are asked to round to the nearest hundredth, which is the same as rounding to two decimal places. As this rounds down, the length of 𝑀𝐶, to the nearest hundredth, is 82.60 meters.

We will now look at a couple of questions where we use the Pythagorean theorem to solve some geometric problems.

Find the area of the square 𝐵𝐸𝐷𝐶.

As 𝐵𝐸𝐷𝐶 is a square, we know that each of the lengths is the same size. The area of any square can be calculated by squaring the length of one of the sides. We also notice that we have a right triangle where two of the lengths are given. The third length is the length 𝑥. We can, therefore, calculate the missing length by using the Pythagorean theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side or hypotenuse.

Substituting in our values, we have 𝑥 squared plus 21 squared is equal to 35 squared. This is because 35 is the hypotenuse. 21 squared is equal to 441. And 35 squared is 1225. We can subtract 441 from both sides, giving us 𝑥 squared is equal to 784. Square rooting both sides of this equation gives us 𝑥 is equal to 28. The length of each side on the square is 28 centimeters.

In this question, there was actually a shortcut we could have used to calculate the length of 𝐵𝐶. One of our Pythagorean triples is three four five. This means that any triangle with sides in this ratio will be a right triangle. The hypotenuse, or longest side of our triangle, was 35 centimeters. And one of the shorter sides was 21 centimeters. Three multiplied by seven is 21, and five multiplied by seven is 35. As four multiplied by seven is equal to 28, the missing length in the triangle was 28 centimeters. This confirms our previous calculation.

We can then calculate the area of the square by squaring 28. As 28 squared is equal to 784, the area of square 𝐵𝐸𝐷𝐶 is 784 square centimeters. Our answer for area will always be in square units.

We will now look at a second geometric problem.

Find the perimeter of 𝐴𝐵𝐶𝐷.

The perimeter of any shape is the distance around the outside. In this case, we would need to add the lengths 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴. Three of these lengths are given to us. And we will call the length 𝐷𝐴 𝑥 centimeters. Substituting in the values we know gives us a perimeter of 20 plus 48 plus 39 plus 𝑥. This simplifies to 107 plus 𝑥.

We notice that our quadrilateral or four-sided shape is split into two right triangles. This means that we could use the Pythagorean theorem to calculate any missing lengths. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side or hypotenuse. In this question, however, there is a quicker method using our knowledge of Pythagorean triples.

Two of these such triples are five 12 13 and three four five. This means that any triangle with three lengths in these ratios will be a right triangle. Let’s begin by considering the orange triangle with lengths 20 centimeters and 48 centimeters and hypotenuse 𝑦. Five multiplied by four is equal to 20, and 12 multiplied by four is equal to 48. This means that we can calculate the length 𝑦 by multiplying 13 by four. This is equal to 52. Therefore, the length of 𝐴𝐶 is 52 centimeters.

In our pink triangle, the two shorter sides have lengths 39 and 52 centimeters. The length of the hypotenuse, or longest side, is 𝑥. Multiplying three and four by 13 gives us 39 and 52, respectively. This means that the longest side 𝑥 will be equal to five multiplied by 13. This is equal to 65. The length of 𝑥 or 𝐴𝐷 is 65 centimeters. Substituting this into our perimeter expression gives us 107 plus 65. 107 plus 65 is equal to 172. We can, therefore, conclude that the perimeter of 𝐴𝐵𝐶𝐷 is 172 centimeters.

Our final question involves applying the converse of the Pythagorean theorem.

The distances between three cities are 77 miles, 36 miles, and 49 miles. Do the positions of these cities form a right triangle?

We can answer this question by considering the Pythagorean theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side or hypotenuse of a right triangle. The converse of the Pythagorean theorem states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right angled.

In this question, we need to consider the sum of the squares of 36 and 49 and see if they’re equal to the square of 77. 77 squared is equal to 5929. 36 squared plus 49 squared is equal to 3697. These two values are not equal. This means that 36 squared plus 49 squared is not equal to 77 squared. We can, therefore, conclude that as the three distances do not satisfy the Pythagorean theorem, the triangle is not a right triangle.

We will now summarize the key points from this video. The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. This is usually written 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse. We can apply this theorem to solve geometric and real-life problems. This includes calculating the length of the hypotenuse or one of the shorter sides.

Our knowledge of Pythagorean triples often provides a shortcut. Examples of Pythagorean triples are three four five and five 12 13. Any triangle with three lengths in these ratios will be a right triangle. We also know that the converse of the Pythagorean theorem is true. If the three lengths of any triangle satisfy 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, then the triangle is a right triangle.