Quiz 7-1 Pythagorean Theorem Special Right Triangles & Geometric Mean (2024)

FAQs

What is the Pythagorean theorem special right triangles? ›

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle. The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a2+b2=c2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

A 45-45-90 triangle is a special type of right triangle, where the ratio of the lengths of the sides of a 45-45-90 triangle is always 1:1:√2, meaning that if one leg is x units long, then the other leg is also x units long, and the hypotenuse is x√2 units long. Created by Sal Khan.

A Pythagorean triple is a special case where the lengths of all the sides of a right triangle are whole numbers. The 3-4-5 triangle is the smallest and best known of the Pythagorean triples.

Key Concept Explanation

In this context, the altitude of a right triangle is the geometric mean of the two segments of the hypotenuse. This relationship is expressed as h=\sqrt{(ac)} , where a and c are the two segments of the hypotenuse.

30-60-90-Triangle Theorem

Statement: The length of the hypotenuse is twice the length of the shortest side and the length of the other side is √3 times the length of the shortest side in a 30-60-90-Triangle.

What Is a “30-60-90” Triangle? A special right triangle with angles 30°, 60°, and 90° is called a 30-60-90 triangle. The angles of a 30-60-90 triangle are in the ratio 1 : 2 : 3. Since 30° is the smallest angle in the triangle, the side opposite to the 30° angle is always the smallest (shortest leg).

To find the hypotenuse, add the squares of the other sides, then take the square root. To find a shorter side, subtract the squares of the other sides, then take the square root.

Answer and Explanation:

The special right triangle that has a hypotenuse of 13 units is the right triangle that has legs of length 5 units and 12 units.

The hypotenuse is always the longest side in a right triangle because it is opposite of the largest angle, the ninety degree angle.

What is the leg rule for similar triangles? ›

Leg-Leg Similarity

If the lengths of the corresponding legs of two right triangles are proportional, then by Side-Angle-Side Similarity the triangles are similar. In the figure, A B P Q = B C Q R . So, Δ A B C ∼ Δ P Q R .

Solving right triangles

Pythagorean theorem: a² + b² = c². Sines: sin A = a/c, sin B = b/c. Cosines: cos A = b/c, cos B = a/c. Tangents: tan A = a/b, tan B = b/a.

The formula for Pythagoras' theorem is a² + b² = c². In this equation, “C” represents the longest side of a right triangle, called the hypotenuse. “A” and “B” represent the other two sides of the triangle.

The Pythagorean Theorem is a special case of the law of cosines, a2 + b2 - 2*a*b*cos (theta) = c2 because cos (theta) = 0 when the angle is a 90 degree or right angle. We also learned that the converse of the Pythagorean Theorem is true as well.