Similar right triangles and the Pythagorean theorem

Warm-up

Three quick questions from earlier lessons.

Discover

The altitude to the hypotenuse of a right triangle creates two smaller right triangles. How are they related to the original?

All three triangles are similar
They are congruent to each other

Key idea

Similar right triangles give the Pythagorean theorem.

If the altitude

h

cuts the hypotenuse

c

into

p

and

q

:

h^2 = pq

; each leg satisfies

a^2 = pc

(with

p

next to

a

),

b^2 = qc

. Adding:

a^2 + b^2 = c(p + q) = c^2

.

Worked example

Segments $p = 4$ , $q = 9$ .

h = \sqrt{36} = 6

h² = pq.

a = \sqrt{52}

c = 13; a² = 4 × 13.

b = \sqrt{117}

b² = 9 × 13.

Your turn, guided

Your turn: legs 6 and 8. The hypotenuse.

10

√(36 + 64).

Practice ★

A right triangle has legs 9 and 12. Find the hypotenuse.
A right triangle has legs 3 and 4. Find the hypotenuse.

The altitude to the hypotenuse of a right triangle cuts the hypotenuse into segments 36 and 49. Find the altitude.
The altitude to the hypotenuse of a right triangle cuts the hypotenuse into segments 9 and 16. Find the altitude.

Watch out!

Practice ★★

In a right triangle, the hypotenuse is 25 and the altitude cuts off a segment of 1 next to leg $a$ . Find $a$ .
In a right triangle, the hypotenuse is 49 and the altitude cuts off a segment of 1 next to leg $a$ . Find $a$ .

Exit ticket

Three questions to check you have got it.

A right triangle has legs 9 and 12. Find the hypotenuse.
A right triangle has legs 3 and 4. Find the hypotenuse.
A right triangle has legs 18 and 24. Find the hypotenuse.

Summary card

Similar right triangles

$h^2 = pq$ , $a^2 = pc$ , $b^2 = qc$ ; hence $a^2 + b^2 = c^2$ .

Watch out: geometric mean, not average.

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