How Right Triangles Demonstrate Similarity

• AA (Angle-Angle) Criterion: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Since right triangles formed by an altitude share a common angle and both have a \(90^\circ\) angle, they’re similar by the AA criterion.

• The altitude is geometrically the mean between the two segments it divides the hypotenuse into.
• Each leg of the large triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

Examples

Practice Questions:

1. In right triangle \(PQR\), if \(PR\) is the hypotenuse and \(PS\) is an altitude dividing \(PR\) into segments of \(3 \text{ cm}\) and \(9 \text{ cm}\), find the length of \(PS\).
2. Given a right triangle \(LMN\) with \(LN\) as the hypotenuse of length \(17 \text{ cm}\) and altitude \(LO\) dividing it into segments of \(8 \text{ cm}\) and \(15 \text{ cm}\), find the lengths of \(LO\) and \(MO\).

1. \(PS = 6 \text{ cm}\)
2. \(LO = 7.2 \text{ cm}\) and \(MO = 9.6 \text{ cm}\)

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