Table of Contents
Right Triangles and the Pythagorean Relationship
In this chapter we focus on a special and very famous relationship that holds only in right triangles: the Pythagorean theorem. You should already know what a triangle is and what a right angle is from earlier geometry chapters; here we concentrate on the specific length relationship.
Consider a right triangle. By convention:
- The side opposite the right angle is called the hypotenuse.
- The other two sides, which form the right angle, are often called legs.
We usually label the hypotenuse as $c$, and the legs as $a$ and $b$.
The Pythagorean theorem says:
$$a^2 + b^2 = c^2$$
where $c$ is the length of the hypotenuse and $a$ and $b$ are the lengths of the other two sides.
In words:
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
This equation is only guaranteed to be true for right triangles.
Using the Pythagorean Theorem to Find a Missing Side
Typically, you know two side lengths of a right triangle and want the third.
Finding the hypotenuse
If you know both legs $a$ and $b$:
- Square both leg lengths: compute $a^2$ and $b^2$.
- Add these squares: compute $a^2 + b^2$.
- Take the square root to get $c$:
$$c = \sqrt{a^2 + b^2}$$
Example: A right triangle has legs $3$ and $4$ units long.
- $3^2 = 9$
- $4^2 = 16$
- $9 + 16 = 25$
- $c = \sqrt{25} = 5$
So the hypotenuse is $5$ units long.
Finding a leg
If you know the hypotenuse $c$ and one leg, say $a$:
- Square both known sides: compute $c^2$ and $a^2$.
- Subtract: $b^2 = c^2 - a^2$.
- Take the square root:
$$b = \sqrt{c^2 - a^2}$$
Example: A right triangle has hypotenuse $13$ and one leg $5$.
- $13^2 = 169$
- $5^2 = 25$
- $b^2 = 169 - 25 = 144$
- $b = \sqrt{144} = 12$
So the other leg is $12$ units long.
Pythagorean Triples
Sometimes all three sides of a right triangle are whole numbers. Sets of three positive integers $(a,b,c)$ that satisfy
$$a^2 + b^2 = c^2$$
are called Pythagorean triples.
Common basic examples are:
- $(3, 4, 5)$ because $3^2 + 4^2 = 9 + 16 = 25 = 5^2$
- $(5, 12, 13)$ because $5^2 + 12^2 = 25 + 144 = 169 = 13^2$
- $(8, 15, 17)$ because $8^2 + 15^2 = 64 + 225 = 289 = 17^2$
You can multiply a triple by any positive number to get another right triangle with the same shape. For example, multiplying $(3,4,5)$ by $2$ gives $(6,8,10)$, which also satisfies the theorem.
Recognizing these triples can make some calculations much quicker, especially in simple geometric problems.
Checking if a Triangle Is Right
If you are given three side lengths and want to know whether they can form a right triangle, you can use the Pythagorean theorem in reverse.
- Identify the longest side; call it $c$.
- Call the other two sides $a$ and $b$.
- Check whether $a^2 + b^2 = c^2$.
- If $a^2 + b^2 = c^2$, then the triangle is right-angled (with $c$ as the hypotenuse).
- If $a^2 + b^2 \neq c^2$, then the triangle is not right-angled.
Example: Sides $7$, $24$, and $25$.
- Longest side: $25$ (so $c = 25$)
- $a = 7$, $b = 24$
- $7^2 = 49$, $24^2 = 576$
- $49 + 576 = 625$
- $25^2 = 625$
Since these match, the triangle with sides $7$, $24$, and $25$ is a right triangle.
Geometric Interpretation with Squares on the Sides
The equation $a^2 + b^2 = c^2$ can be visualized using areas.
Imagine you draw a square on each side of a right triangle:
- On the side of length $a$, you draw a square with area $a^2$.
- On the side of length $b$, you draw a square with area $b^2$.
- On the hypotenuse of length $c$, you draw a square with area $c^2$.
The Pythagorean theorem says:
$$\text{Area on side } a \;+\; \text{Area on side } b \;=\; \text{Area on hypotenuse}$$
That is,
$$a^2 + b^2 = c^2$$
So the combined area of the two smaller squares equals the area of the largest square. Many geometric proofs of the theorem are based on rearranging these areas.
Distance in the Plane (Preview Idea)
The Pythagorean theorem underlies the idea of distance between two points in a flat, coordinate plane.
If you have two points $(x_1,y_1)$ and $(x_2,y_2)$, and you draw a right triangle whose legs are horizontal and vertical:
- One leg has length $|x_2 - x_1|$.
- The other leg has length $|y_2 - y_1|$.
By the Pythagorean theorem, the distance $d$ between the points is
$$d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$
so
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
This will be developed more fully when working with the coordinate plane and analytic geometry.
Practical Uses of the Pythagorean Theorem
The Pythagorean theorem is frequently used in simple real-world length problems involving right angles, such as:
- Measuring diagonals: Finding the diagonal of a rectangle with side lengths $a$ and $b$; the diagonal is the hypotenuse:
$$\text{Diagonal} = \sqrt{a^2 + b^2}.$$ - Ladders and walls: A ladder leaning against a wall, forming a right triangle with the ground and the wall. Knowing two lengths lets you find the third.
- Screen sizes: The “size” of a rectangular screen (like $13$-inch or $24$-inch) is the length of the diagonal, which relates to the width and height via the theorem.
In each of these, the key is to recognize a right angle and then apply $a^2 + b^2 = c^2$ with appropriate labels.
Summary of Key Points
- The Pythagorean theorem applies only to right triangles.
- If $a$ and $b$ are the legs and $c$ is the hypotenuse:
$$a^2 + b^2 = c^2.$$ - You can solve for:
- The hypotenuse: $c = \sqrt{a^2 + b^2}$.
- A leg: $a = \sqrt{c^2 - b^2}$ or $b = \sqrt{c^2 - a^2}$.
- Pythagorean triples are integer solutions to $a^2 + b^2 = c^2$.
- The theorem can be used to check whether a triangle is right and to relate areas of squares built on each side.
- It is a foundational tool for distance and length problems throughout geometry and beyond.