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G-SRT.4 Core M2-049-R08-V04

Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.

Derive the Pythagorean Theorem from altitude similarity

Problem

Complete the altitude-similarity proof from \(XZ^{2}~=~XY\cdot~XW\) and \(YZ^{2}~=~XY\cdot~YW\). Add the equations, factor \(XY\), apply segment addition \(XW~+~YW~=~XY\), and state the resulting Pythagorean identity.

Unworked right triangle XYZ with right angle at Z and altitude ZW perpendicular to hypotenuse XY at W; hypotenuse pieces XW and YW and the supplied relations XZ²=XY·XW and YZ²=XY·YW are shown, while their summed Pythagorean conclusion is withheld. Open full size
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Four variants of this problem type
Curriculum context
Course
Math II
Standard
G-SRT.4
Category
Geometry
Domain
Similarity, Right Triangles, and Trigonometry
Objective
Prove triangle-similarity theorems, including proportional segments and the Pythagorean Theorem via similarity.
Problem type
Derive the Pythagorean Theorem from altitude similarity