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G-SRT.4 Core M2-049-R08-V03

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

Derive the Pythagorean Theorem from altitude similarity

Problem

In the right-triangle altitude configuration, \(GH~=~c\) is divided into \(GK~=~p\) and \(KH~=~q\). Use the \(\text{two}~\text{small}-\triangle~\text{similarities}\) to derive \(GJ^{2}~=~\text{cp}\) and \(HJ^{2}~=~\text{cq}\), add the identities, and use \(p~+~q~=~c\) to obtain the Pythagorean relationship.

Unworked right triangle GHJ with right angle at J and altitude JK perpendicular to hypotenuse GH at K; GH=c, GK=p, and KH=q are supplied, while leg equations and the Pythagorean identity are 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