1.1: The Pythagorean Theorem

Learning Objectives

• Use the Pythagorean Theorem to determine if a triangle is a right triangle.
• Use the Pythagorean Theorem to determine the length of one side of a right triangle.
• Use the distance formula to determine the distance between two points on the coordinate plane.

Recall the following definitions from elementary geometry:

1. An angle is acute if it is between $0°$ and $90°$.
2. An angle is a right angle if it equals $90°$.
3. An angle is obtuse if it is between $90°$ and $180°$.
4. An angle is a straight angle if it equals $180°$.

Figure 1.1.1 Types of angles

In elementary geometry, angles are always considered to be positive and not larger than $360^\circ$. For now we will only consider such angles. The following definitions will be used throughout the text:

1. Two acute angles are complementary if their sum equals $90^◦$. In other words, if $0^◦ ≤ ∠ A , ∠B ≤ 90^◦ \text{ then }∠ A \text{ and }∠B$ are complementary if $∠ A +∠B = 90^◦$.
2. Two angles between $0^◦ \text{ and }180^◦$ are supplementary if their sum equals $180^◦$. In other words, if $0^◦ ≤ ∠ A , ∠B ≤ 180^◦ \text{ then }∠ A \text{ and }∠B$ are supplementary if $∠ A +∠B = 180^◦$.
3. Two angles between $0^◦ \text{ and }360^◦$ are conjugate (or explementary) if their sum equals $360^◦$. In other words, if $0^◦ ≤ ∠ A , ∠B ≤ 360^◦ \text{ then }∠ A \text{ and }∠B\text{ are conjugate if }∠ A+∠B = 360^◦$.

Figure 1.1.2 Types of pairs of angles

Instead of using the angle notation $∠ A$ to denote an angle, we will sometimes use just a capital letter by itself (e.g. $A, B, C$) or a lowercase variable name (e.g. $x, y, t$). It is also common to use letters (either uppercase or lowercase) from the Greek alphabet, shown in the table below, to represent angles:

Table 1.1 The Greek alphabet

In elementary geometry you learned that the sum of the angles in a triangle equals $180^◦$, and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is $90^◦$ and the other two angles are acute angles whose sum is $90^◦$ (i.e. the other two angles are complementary angles).

$$\nonumber α + 3α + α = 180^◦ ⇒ 5α = 180^◦ ⇒ α = 36^◦ ⇒ \fbox{\(X = 36^◦ ,\, Y = 3×36^◦ = 108^◦ ,\, Z = 36^◦\)}$$

QED

Figure 1.1.3

By knowing the lengths of two sides of a right triangle, the length of the third side can be determined by using the Pythagorean Theorem:

$a^2+b^2=c^2$

The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its legs.

Figure 1.1.4 Similar triangles $△ ABC, △CBD, △ ACD$

Recall that triangles are similar if their corresponding angles are equal, and that similarity implies that corresponding sides are proportional. Thus, since $\triangle\,ABC$ is similar to $\triangle\,CBD$, by proportionality of corresponding sides we see that

$$\nonumber \overline{AB}~\text{is to}~\overline{CB}~\text{(hypotenuses)}\text{ as } \overline{BC}~\text{is to}~\overline{BD}~\text{(vertical legs)} \quad\Rightarrow\quad \frac{c}{a} ~=~ \frac{a}{d} \quad\Rightarrow\quad cd ~=~ a^2 ~.$$

Since $\triangle\,ABC$ is similar to $\triangle\,ACD$, comparing horizontal legs and hypotenuses gives

$$\nonumber \frac{b}{c-d} ~=~ \frac{c}{b} \quad\Rightarrow\quad b^2 ~=~ c^2 ~-~ cd ~=~ c ^2 ~-~ a^2 \quad\Rightarrow\quad a^2 ~+~ b^2 ~=~ c^2 ~. \textbf{QED}$$

Note: The symbols $\perp$ and $\sim$ denote perpendicularity and similarity, respectively. For example, in the above proof we had $\,\overline{CD} \perp \overline{AB}\,$ and $\,\triangle\,ABC \sim \triangle\,CBD \sim \triangle\,ACD$.

For triangle $\triangle\,ABC$, the Pythagorean Theorem says that

$$\nonumber a^2 ~+~ 4^2 ~=~ 5^2 \quad\Rightarrow\quad a^2 ~=~ 25 ~-~ 16 ~=~ 9 \quad\Rightarrow\quad \fbox{\(a ~=~ 3\)} ~.$$

For triangle $\triangle\,DEF$, the Pythagorean Theorem says that

$$\nonumber e^2 ~+~ 1^2 ~=~ 2^2 \quad\Rightarrow\quad e^2 ~=~ 4 ~-~ 1 ~=~ 3 \quad\Rightarrow\quad \fbox{$e ~=~ \sqrt{3}$} ~.$$

For triangle $\triangle\,XYZ$, the Pythagorean Theorem says that

$$\nonumber 1^2 ~+~ 1^2 ~=~ z^2 \quad\Rightarrow\quad z^2 ~=~ 2 \quad\Rightarrow\quad \fbox{$z ~=~ \sqrt{2}$} ~.$$

Let $h$ be the height at which the ladder touches the wall. We can assume that the ground makes a right angle with the wall, as in the picture on the right. Then we see that the ladder, ground, and wall form a right triangle with a hypotenuse of length 17 ft (the length of the ladder) and legs with lengths 8 ft and $h$ ft. So by the Pythagorean Theorem, we have

$$\nonumber h^2 ~+~ 8^2 ~=~ 17^2 \quad\Rightarrow\quad h^2 ~=~ 289 ~-~ 64 ~=~ 225 \quad\Rightarrow\quad \fbox{$h ~=~ 15 ~\text{ft}$} ~.$$

Determining the Distance Using the Pythagorean Theorem

You can use the Pythagorean Theorem is to find the distance between two points.

Consider the points (-1, 6) and (5, -3). If we plot these points on a grid and connect them, they make a diagonal line. Draw a vertical line down from (-1, 6) and a horizontal line to the left of (5, -3) to make a right triangle.

Figure $\PageIndex{5}$

Now we can find the distance between these two points by using the vertical and horizontal distances that we determined from the graph.

$\begin{aligned} 9^2+(−6)^2 &=d^2 \\ 81+36 &=d^2 \\ 117 &=d^2\\ \sqrt{117}&=d \\ 3\sqrt{13}&=d\end{aligned}$

Notice, that the x−values were subtracted from each other to find the horizontal distance and the y−values were subtracted from each other to find the vertical distance. If this process is generalized for two points $(x_1, y_1)$and $(x_2, y_2)$, the Distance Formula is derived.

$(x_1−x_2)^2+(y_1−y_2)^2=d^2$

Figure $\PageIndex{6}$

This is the Pythagorean Theorem with the vertical and horizontal differences between $(x_1, y_1)$ and $(x_2, y_2)$. Taking the square root of both sides will solve the right hand side for d, the distance.

$\sqrt{(x_1−x_2)^2+(y_1−y_2)^2}=d$

This is the Distance Formula.

Contributors and Attributions

• Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2.