Using the Pythagorean Theorem to Find Distance
Using the Pythagorean Theorem to Find Distance is a Grade 8 math skill from Reveal Math, Course 3, Module 7: Triangles and the Pythagorean Theorem. By drawing a right triangle between two points on a coordinate plane, students can use the formula a^2 + b^2 = c^2—where a is the horizontal distance, b is the vertical distance, and c is the straight-line distance—to find the exact length between any two points. For example, two points with a horizontal distance of 3 and a vertical distance of 4 are exactly 5 units apart (since 9 + 16 = 25, and the square root of 25 is 5). This technique is foundational in 8th grade geometry and leads directly to the Distance Formula studied in high school algebra.
Key Concepts
When finding the distance between two points on a coordinate plane, you can use the Pythagorean Theorem: $$a^2 + b^2 = c^2$$ where $a$ is the horizontal distance, $b$ is the vertical distance, and $c$ is the straight line distance (the hypotenuse) between the two points. To find the distance $c$, calculate: $$c = \sqrt{a^2 + b^2}$$.
Common Questions
How do you use the Pythagorean Theorem to find distance between two points?
Draw a right triangle using the two points as endpoints of the hypotenuse. The horizontal distance is leg a, and the vertical distance is leg b. Then calculate c = sqrt(a squared + b squared) to find the straight-line distance.
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a squared plus b squared equals c squared. It is used to find missing side lengths in right triangles.
How do you find horizontal and vertical distance between two points?
The horizontal distance is the absolute difference between the x-coordinates of the two points. The vertical distance is the absolute difference between the y-coordinates. These form the legs of the right triangle.
Why does the Pythagorean Theorem work for finding distance on a coordinate plane?
When you connect two points on a coordinate plane, the horizontal and vertical differences form the legs of a right triangle. The straight-line distance between the points is the hypotenuse, so the Pythagorean Theorem applies directly.
When do 8th graders learn to apply the Pythagorean Theorem to find distance?
In Grade 8 Reveal Math Course 3, applying the Pythagorean Theorem to coordinate plane distances is covered in Module 7: Triangles and the Pythagorean Theorem.
What Pythagorean triple helps you quickly find distance without a calculator?
The 3-4-5 triple is the most common. If two points have a horizontal distance of 3 and vertical distance of 4, the straight-line distance is exactly 5, without needing a calculator. Other triples include 5-12-13 and 8-15-17.