Pythagorean Theorem and Special Right Triangles

Front

What is the Pythagorean Theorem?

Back

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It can be expressed as: c² = a² + b².

Front

What are the side lengths of a 30-60-90 triangle?

Back

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1 : √3 : 2. The side opposite the 30° angle is the shortest (1), the side opposite the 60° angle is √3, and the hypotenuse is 2.

Front

What are the side lengths of a 45-45-90 triangle?

Back

In a 45-45-90 triangle, the lengths of the legs are equal, and the hypotenuse is √2 times the length of each leg. If each leg is 'a', then the hypotenuse is a√2.

Front

How do you find the length of a side in a right triangle using the Pythagorean Theorem?

Back

To find the length of a side, rearrange the Pythagorean Theorem: a² = c² - b² (to find a) or b² = c² - a² (to find b). Then take the square root of the result.

Front

What is the relationship between the angles and sides in a triangle?

Back

In any triangle, the sum of the interior angles is always 180 degrees. The longest side is opposite the largest angle.

Front

How can you determine if a triangle is acute, obtuse, or right using side lengths?

Back

Use the Pythagorean Theorem: If a² + b² = c², the triangle is right; if a² + b² > c², it is acute; if a² + b² < c², it is obtuse.

Front

What is the formula to calculate the diagonal of a rectangle?

Back

The diagonal (d) of a rectangle can be calculated using the Pythagorean Theorem: d = √(length² + width²).

Front

If a triangle has sides of lengths 11, 12, and 15, how can you classify it?

Back

To classify the triangle, check if 11² + 12² > 15². Since 121 + 144 > 225, the triangle is acute.

Front

What is the height of a right triangle if the base is 5 and the hypotenuse is 13?

Back

Using the Pythagorean Theorem: height² = 13² - 5²; height² = 169 - 25; height = √144 = 12.

Front

What is the length of the hypotenuse in a 45-45-90 triangle with legs of length 7?

Back

The hypotenuse is 7√2.