How Right Triangles Demonstrate Similarity

Two right triangles are similar when they share an acute angle, because the right angles already match and the third angle then follows. That single shared angle is why the altitude to the hypotenuse creates three similar triangles.

• 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.

Worked examples: similarity in right triangles

Practice questions on right triangle similarity

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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