PYTHAGOREAN THEOREM & BASICS OF TRIANGLES

Basics of Triangles

A triangle is a geometric shape with three sides. Triangles can be classified based on the lengths of their sides and the measures of their angles.

Triangles Classified by Sides

• Equilateral Triangle: All three sides are equal in length. All angles are also equal (each 60°).

• Isosceles Triangle: Two sides are equal in length, and the third is different. The angles opposite the equal sides are also equal.

• Scalene Triangle: All three sides are of different lengths. All angles are also different.

Triangles Classified by Angles

• Acute Triangle: All three angles are less than 90°.

• Obtuse Triangle: One angle is greater than 90°.

• Right Triangle: One angle is exactly 90°.

When two triangles have the same side lengths, they are congruent. The classification by angles and sides helps in identifying triangle properties and solving geometric problems.

Example

For each triangle, find the missing angle or side as required. (See diagram for visual reference.)

Right Triangles & Pythagorean Theorem

A right triangle is a triangle with one angle equal to 90°. The two sides that form the right angle are called the legs, and the side opposite the right angle is the hypotenuse.

The Pythagorean Theorem relates the lengths of the sides in a right triangle:

• a and b are the lengths of the legs.

• c is the length of the hypotenuse (the side opposite the right angle).

To find a missing side in a right triangle, substitute the known values into the Pythagorean Theorem and solve for the unknown.

Example

Find the missing side of each right triangle below:

• If the legs are 3 and 4, the hypotenuse is found by , so , thus .

• If one leg is 6 and the hypotenuse is y, then .

Practice: Use the Pythagorean Theorem to calculate the missing side for given right triangles.