Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 12: Right Triangles and Quadrilaterals

Lesson 1: The Pythagorean Theorem

In this Grade 4 AMC Math lesson from The Art of Problem Solving: Prealgebra, students explore the Pythagorean Theorem, learning how the relationship a² + b² = c² connects the legs and hypotenuse of any right triangle. The lesson walks through a geometric proof using four congruent right triangles arranged around a square, guiding students to derive the theorem by comparing areas. Students also practice identifying Pythagorean triples and applying the theorem to find missing side lengths while avoiding common errors like forgetting to take the square root.

Section 1

Pythagorean Theorem: Formula and Area Relationship

Property

For a right triangle whose leg lengths are aa and bb and whose hypotenuse is of length cc, the relationship between the side lengths is given by the formula:

a2+b2=c2a^2 + b^2 = c^2
This result can be shown by observing that the area of a tilted square with side cc inside an (a+b)(a+b)-sided square is equal to the sum of the areas of an aa-sided square and a bb-sided square.

Examples

  • A right triangle has legs of length a=5a=5 and b=12b=12. To find the hypotenuse cc, we use c2=52+122=25+144=169c^2 = 5^2 + 12^2 = 25 + 144 = 169. So, c=169=13c = \sqrt{169} = 13.
  • A right triangle has a hypotenuse of length c=10c=10 and one leg of length a=6a=6. To find the other leg bb, we use b2=c2−a2=102−62=100−36=64b^2 = c^2 - a^2 = 10^2 - 6^2 = 100 - 36 = 64. So, b=64=8b = \sqrt{64} = 8.
  • The legs of a right triangle are both 3 units long. The hypotenuse cc is found with c2=32+32=9+9=18c^2 = 3^2 + 3^2 = 9 + 9 = 18. The length of the hypotenuse is c=18c = \sqrt{18}.

Explanation

This famous theorem is a secret code for right triangles! It connects the lengths of the two shorter sides (legs) to the longest side (hypotenuse). If you know the lengths of any two sides, you can always find the third.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Pythagorean Theorem: Formula and Area Relationship

Property

For a right triangle whose leg lengths are aa and bb and whose hypotenuse is of length cc, the relationship between the side lengths is given by the formula:

a2+b2=c2a^2 + b^2 = c^2
This result can be shown by observing that the area of a tilted square with side cc inside an (a+b)(a+b)-sided square is equal to the sum of the areas of an aa-sided square and a bb-sided square.

Examples

  • A right triangle has legs of length a=5a=5 and b=12b=12. To find the hypotenuse cc, we use c2=52+122=25+144=169c^2 = 5^2 + 12^2 = 25 + 144 = 169. So, c=169=13c = \sqrt{169} = 13.
  • A right triangle has a hypotenuse of length c=10c=10 and one leg of length a=6a=6. To find the other leg bb, we use b2=c2−a2=102−62=100−36=64b^2 = c^2 - a^2 = 10^2 - 6^2 = 100 - 36 = 64. So, b=64=8b = \sqrt{64} = 8.
  • The legs of a right triangle are both 3 units long. The hypotenuse cc is found with c2=32+32=9+9=18c^2 = 3^2 + 3^2 = 9 + 9 = 18. The length of the hypotenuse is c=18c = \sqrt{18}.

Explanation

This famous theorem is a secret code for right triangles! It connects the lengths of the two shorter sides (legs) to the longest side (hypotenuse). If you know the lengths of any two sides, you can always find the third.