When a triangle is inscribed in a semicircle where a...

Name: Problem 97, Chapter 8: Intermediate Algebra for College Students
Uploaded: 2021-10-31T08:08:17-08:00
Duration: 2 min 16 s
Channel: Jay Patel
Description: When a triangle is inscribed in a semicircle where a diameter of the circle is a side of the triangle, the triangle formed is always a right triangle. If an isosceles triangle (two equal sides) is inscribed in a semicircle of radius 10 inches, determine the length of the other two sides of the triangle.

When a triangle is inscribed in a semicircle where a diameter of the circle is a side of the triangle, the triangle formed is always a right triangle. If an isosceles triangle (two equal sides) is inscribed in a semicircle of radius 10 inches, determine the length of the other two sides of the triangle.

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Key Concepts

Pythagorean Theorem

The Pythagorean theorem establishes that in a right triangle the sum of the squares of the two legs equals the square of the hypotenuse. This theorem is essential to calculate the unknown side lengths in any right triangle, including the determination of the legs of an isosceles right triangle when the hypotenuse is known.

Isosceles Triangle Properties

An isosceles triangle has two sides of equal length and, consequently, two equal angles. When the isosceles triangle is also a right triangle, these properties combine to form an isosceles right triangle, leading to a specific relationship among the side lengths where the equal sides are related to the hypotenuse by a factor of ?2.

Right Triangle Properties

A right triangle, characterized by one 90-degree angle, obeys specific geometric rules including the Pythagorean theorem. These properties allow one to relate the lengths of the sides in any right-angled triangle, which is crucial for solving problems involving spatial relationships in circles.

Thales' Theorem

This theorem states that any triangle inscribed in a semicircle, where one side is the diameter, is a right triangle. It is a fundamental property of circles and inscribed angles, and it guarantees that the angle opposite the diameter in such triangles is always 90 degrees.

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