Finding a shorter side and applying Pythagoras

Finding a shorter side of a right-angled triangle with a² = c² − b², and solving real problems with ladders, diagonals and distances.

A shorter side

Example 1●

A right-angled triangle has hypotenuse $17$ cm and one side $8$ cm. Find the third side.

$a^2 = 289 - 64 = 225$

$a = 15$ cm

Example 2: a ladder●●

A $6$ m ladder leans against a wall with its foot $1.8$ m from the wall. How high up the wall does it reach, to 2 decimal places?

The ladder is the hypotenuse: $h^2 = 6^2 - 1.8^2 = 36 - 3.24 = 32.76$.

$h = \sqrt{32.76} \approx 5.72$ m

Example 3: a shortcut●●

A rectangular park is $120$ m by $50$ m. How much shorter is it to walk along the diagonal than around two sides?

Diagonal: $\sqrt{120^2 + 50^2} = \sqrt{16\,900} = 130$ m.

Two sides: $170$ m. Saving: $40$ m.

Flashcards

Shorter side formula

$a^2 = c^2 - b^2$

Ladder problems

The ladder is the hypotenuse.

Hypotenuse 13, side 5

Other side $12$

Practice