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G-SRT.4 Core M2-049-R08-V02

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

Derive the Pythagorean Theorem from altitude similarity

Problem

In right triangle \(PQR\), the altitude to hypotenuse \(PQ~=~15\) divides it into \(PS~=~27/5\) and \(SQ~=~48/5\). Use altitude similarity to derive and evaluate \(PR^{2}~=~PQ\cdot~PS\) and \(QR^{2}~=~PQ\cdot~SQ\), then add them to recover \(PQ^{2}\).

Neutral right-triangle altitude reference listing PQ 15, PS 27 over 5, and SQ 48 over 5, without solved leg projections or computed lengths. 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