Angles in inscribed right triangles

Learning Objectives Identify when a triangle inscribed in a circle is a right triangle. State and explain Thales' Theorem and its converse. Prove that an angle inscribed in a semicircle is a right angle. Calculate missing angles in an inscribed right triangle using circle properties and the angle sum of a triangle. Apply the Pythagorean theorem to find unknown side lengths of inscribed right triangles. Use trigonometric ratios to solve for side lengths and angles in inscribed right triangles. Determine the radius or diameter of a circle given an inscribed right triangle. Ever wondered how to draw a perfect 90° corner using just a circle and a straight edge? 📐 This special relationship is a cornerstone of geometry! This tutorial explores the powerful connection betwe...

TermDefinitionExample Inscribed TriangleA triangle whose three vertices all lie on the circumference of a circle.If points A, B, and C are all on the edge of circle O, then triangle ABC is an inscribed triangle. DiameterA straight line segment that passes through the center of a circle and whose endpoints lie on the circle. It is the longest chord of a circle.In a circle with center O, if line segment AC passes through O, then AC is a diameter. SemicircleAn arc of a circle that measures 180°, or exactly half of a circle. A diameter divides a circle into two semicircles.The arc from point A to point C, passing through point B, forms a semicircle if AC is the diameter. Inscribed AngleAn angle formed by two chords in a circle that have a common endpoint on the circle's circumference.If...

Thales' Theorem If A, B, and C are distinct points on a circle where the line segment AC is a diameter, then the angle ∠ABC is a right angle (90°). Use this rule to identify a 90° angle in a triangle whenever you see one of its sides is the diameter of the circle it's inscribed in. This is the foundation for solving most problems on this topic. Converse of Thales' Theorem If a right triangle is inscribed in a circle, then its hypotenuse must be a diameter of the circle. Use this rule when you know a triangle inscribed in a circle has a 90° angle. This allows you to identify the hypotenuse as the circle's diameter, which is useful for finding the circle's radius or center. Pythagorean Theorem a^2 + b^2 = c^2 In any right triangle, the square o...