Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 4: Real Numbers and the Pythagorean Theorem

Lesson 3: The Pythagorean Theorem

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, students explore the Pythagorean Theorem, learning how the relationship a² + b² = c² connects the leg lengths and hypotenuse of a right triangle. Through hands-on activities, students construct a geometric proof using squares drawn on each side of a right triangle, then apply the theorem to find missing side lengths. The lesson also extends to real-life problems, including three-dimensional scenarios such as calculating the length of a guy wire attached to a telephone pole.

Section 1

Pythagorean Theorem

Property

A triangle in which one of the angles is a right angle, or 90∘90^\circ, is called a right triangle. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are the legs.

If cc stands for the length of the hypotenuse, and the lengths of the two legs are aa and bb, then:

a2+b2=c2a^2 + b^2 = c^2

In words: In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

Examples

  • A right triangle has legs of length 3 cm and 4 cm. To find the hypotenuse cc, we use 32+42=c23^2 + 4^2 = c^2. This gives 9+16=259 + 16 = 25, so c2=25c^2=25, and the hypotenuse is 5 cm.

Section 2

Applying the Pythagorean Theorem to Find a Missing Side

Property

The Pythagorean theorem states that for a right triangle with legs of length aa and bb and a hypotenuse of length cc, the relationship is a2+b2=c2a^2 + b^2 = c^2. This theorem is used to find an unknown side length in any right triangle when the other two side lengths are known.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Pythagorean Theorem

Property

A triangle in which one of the angles is a right angle, or 90∘90^\circ, is called a right triangle. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are the legs.

If cc stands for the length of the hypotenuse, and the lengths of the two legs are aa and bb, then:

a2+b2=c2a^2 + b^2 = c^2

In words: In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

Examples

  • A right triangle has legs of length 3 cm and 4 cm. To find the hypotenuse cc, we use 32+42=c23^2 + 4^2 = c^2. This gives 9+16=259 + 16 = 25, so c2=25c^2=25, and the hypotenuse is 5 cm.

Section 2

Applying the Pythagorean Theorem to Find a Missing Side

Property

The Pythagorean theorem states that for a right triangle with legs of length aa and bb and a hypotenuse of length cc, the relationship is a2+b2=c2a^2 + b^2 = c^2. This theorem is used to find an unknown side length in any right triangle when the other two side lengths are known.

Examples