Explainer: The Pythagorean Theorem

In this explainer, we will learn how to use the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and its area.

The Pythagorean theorem gives a relationship between the sides of any right triangle.

Before we start, let’s remember what a right triangle is and how to recognize its hypotenuse.

Definition: Useful Language

A right triangle is a triangle that has one right angle and always one longest side. This longest side is always the side that is opposite the right angle, while the other sides, called the legs, form the right angle. The longest side is called the hypotenuse.

Let’s start by considering an isosceles right triangle, 𝑇, shown in the figure. Squares have been added to each side of 𝑇.

As 𝑇 is isosceles, we see that the squares drawn at the legs are made each of two 𝑇s. And we see also that four 𝑇s fit in the bigger square. We deduce from this that the area of the bigger square, 𝐴, is equal to the sum of the area of the two other squares, 𝐴 and 𝐴. We can write 𝐴+𝐴=𝐴.

Let π‘Ž and 𝑏 be the sides of the legs of the triangle (so, in this special case, π‘Ž=𝑏) and 𝑐 be the side of the hypotenuse. Therefore, 𝐴=π‘ŽοŠ§οŠ¨, 𝐴=π‘οŠ¨οŠ¨, and 𝐴=π‘οŠ©οŠ¨, and by substituting these into the equation, we find π‘Ž+𝑏=𝑐.

This result can be generalized to any right triangle: this is the Pythagorean theorem.

The Pythagorean Theorem

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 β€œπ‘β€ and the two shorter sides are labeled β€œπ‘Žβ€ and β€œπ‘,” then the Pythagorean theorem states that π‘Ž+𝑏=𝑐.

There are many proofs of the Pythagorean theorem. We are going to look at one of them.

Let’s consider a square of length π‘Ž and another square of length 𝑏 that are placed in two opposite corners of a square of length π‘Ž+𝑏 as shown in the diagram below.

By expanding (π‘Ž+𝑏), we can find the area of the two little squares (shaded in blue and green) and of the yellow rectangles. We find (π‘Ž+𝑏)=π‘Ž+2π‘Žπ‘+π‘οŠ¨οŠ¨οŠ¨. This can be found as well by considering that the big square of length π‘Ž+𝑏 is made of a square of area π‘ŽοŠ¨, another square of area π‘οŠ¨, and two rectangles of area π‘Žπ‘.

Now, the blue square and the green square are removed from the big square, and the yellow rectangles are split along one of their diagonals, creating four congruent right triangles. They are then placed in the corners of the big square, as shown in the figure.

As the four yellow triangles are congruent, the four sides of the white shape at the center of the big square are of equal length. (This is the hypotenuse of the yellow right triangles.) Also, the angle of the white shape and the two non-right angles of the right form a straight line. As the measures of the two non-right angles of a right triangle add up to 90∘, the angle of the white shape is 90∘. Therefore, the white shape is a square.

Let 𝑐 be the length of the white square’s side (and of the hypotenuse of the yellow triangles). Using the fact that the big square is made of the white square and the four yellow right triangles, we find that the area of the big square is 𝑐+4Γ—π‘Žπ‘2, that is, 𝑐+2π‘Žπ‘οŠ¨.

Since the big squares in both diagrams are congruent (with side π‘Ž+𝑏), we find that π‘Ž+2π‘Žπ‘+𝑏=𝑐+2π‘Žπ‘οŠ¨οŠ¨οŠ¨, and so π‘Ž+𝑏=𝑐.

This is a geometric proof of the Pythagorean theorem.

Now that we know the Pythagorean theorem, let’s see how to use it to solve some questions.

Example 1: Using the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

Find π‘₯ in the right triangle shown.

Answer

  1. Identify the hypotenuse and the legs in the right triangle. The right angle is ∠𝐹. The legs form the right angle, so they are the sides 𝐹𝐷 and 𝐹𝐸. The hypotenuse is the side opposite ∠𝐹; it is 𝐸𝐷.
  2. With π‘Ž and 𝑏 the legs of the right triangle and 𝑐 the hypotenuse, the Pythagorean theorem states that π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨.
  3. Substitute π‘Ž, 𝑏, and 𝑐 with the actual values given on the diagram into the equation: 5+12=π‘₯.
  4. Solve for π‘₯: 25+144=π‘₯169=π‘₯.

As π‘₯ is a length, it is positive; so π‘₯=√169=13.

Example 2: Using the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

Given that 𝐴𝐡𝐢 is a right angle triangle, where 𝐢𝐡=8cm and 𝐴𝐢=10cm, determine the length of 𝐴𝐡.

Answer

  1. Identify the hypotenuse and the legs in the right triangle. The right angle is ∠𝐡. The legs form the right angle, so they are the sides 𝐡𝐴 and 𝐡𝐢. The hypotenuse is the side opposite ∠𝐡; it is 𝐴𝐢.
  2. With π‘Ž and 𝑏 the legs of the right triangle and 𝑐 the hypotenuse, the Pythagorean theorem states that π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨.
  3. Let 𝐴𝐡 be π‘₯. Substitute π‘Ž, 𝑏, and 𝑐 with their actual values into the equation: 8+π‘₯=10.
  4. Solve for π‘₯: 64+π‘₯=100.
    By subtracting 64 from each side of the equation, π‘₯=100βˆ’64π‘₯=36.

As π‘₯ is a length, it is positive; so π‘₯=√36=6.cm

Let’s summarize how to use the Pythagorean theorem to find an unknown side of a right triangle.

How to Use the Pythagorean Theorem to Find an Unknown Side of a Right Triangle

  1. Identify the hypotenuse and the legs in the right triangle.
  2. With π‘Ž and 𝑏 the legs of the right triangle and 𝑐 the hypotenuse, write the Pythagorean theorem: π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨.
  3. Substitute π‘Ž, 𝑏, and 𝑐 with their actual values, using π‘₯ for the unknown side, into the above equation.
  4. Solve for π‘₯.

The following example is a slightly more complex question where we need to use the Pythagorean theorem.

Example 3: Using the Pythagorean Theorem in Multistep Problems

Find the perimeter of 𝐴𝐡𝐢𝐷.

Answer

  1. Identify the question: find the perimeter of 𝐴𝐡𝐢𝐷.
  2. Identify what is given:
    1. We have 𝐴𝐡,𝐡𝐢, and 𝐢𝐷.
    2. △𝐴𝐡𝐢 is a right triangle at 𝐡, and △𝐴𝐢𝐷 is a right triangle at 𝐢.
  3. Identify a solving strategy:
    1. We want to find 𝐴𝐷 to be able to calculate the perimeter of 𝐴𝐡𝐢𝐷. 𝐴𝐷 is the hypotenuse of 𝐴𝐢𝐷, but we do not know 𝐴𝐢. However, 𝐴𝐢 is the hypotenuse of 𝐴𝐡𝐢 where we know both 𝐴𝐡 and 𝐡𝐢. Therefore, we will apply the Pythagorean theorem first in triangle 𝐴𝐡𝐢 to find 𝐴𝐢 and then in triangle 𝐴𝐢𝐷 to find 𝐴𝐷.
    2. To answer the question, we will work out 𝐴𝐡+𝐡𝐢+𝐢𝐷+𝐴𝐷.
  4. Implement the strategy:
    1. In triangle 𝐴𝐡𝐢, 𝐴𝐢 is the hypotenuse. The Pythagorean theorem states that π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨ in a right triangle where π‘Ž and 𝑏 are the legs and 𝑐 is the hypotenuse. If we substitute the real values for π‘Ž and 𝑏 and let π‘₯=𝐴𝐢, we get 20+48=π‘₯400+2,304=π‘₯2,704=π‘₯. As π‘₯ is a length, it is positive; so π‘₯=√2,704=52.cm
    2. In 𝐴𝐢𝐷, 𝐴𝐷 is the hypotenuse. The Pythagorean theorem states that π‘Ž+𝑏=π‘οŠ¨οŠ¨οŠ¨ in a right triangle where π‘Ž and 𝑏 are the legs and 𝑐 is the hypotenuse. If we substitute the real values for π‘Ž and 𝑏 and let 𝑦=𝐴𝐷, we get 52+39=𝑦2,704+1,521=𝑦4,225=𝑦. As 𝑦 is a length, it is positive; so 𝑦=√4,225=65.cm
    3. We work out the perimeter of 𝐴𝐡𝐢𝐷: 𝐴𝐡+𝐡𝐢+𝐢𝐷+𝐴𝐷=20+48+39+65=172.cm

The perimeter of 𝐴𝐡𝐢𝐷 is 172 cm.

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