In quadrilateral ABCD, ∠ B is a right angle, diagonal AC is perpendicular to CD, AB = 18, BC = 21, and CD = 14. Find the perimeter of ABCD.

2006 AIME I Problem 1

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In quadrilateral

ABCD,

\angle B

is a right angle, diagonal

\overline{AC}

is perpendicular to

\overline{CD},

AB = 18,

BC = 21,

and

CD = 14.

Find the perimeter of

ABCD.

Answer

Diagonal

\overline{AC}

splits the quadrilateral into two right triangles that share

AC

Apply the Pythagorean theorem twice:

DA^2 = (AB^2 + BC^2) + CD^2

Triangle

ABC

is right-angled at

B,

AC^2 = 18^2 + 21^2 = 765.

Triangle

ACD

is right-angled at

C,

DA^2 = AC^2 + CD^2

= 765 + 196 = 961,

giving

DA = 31.

The perimeter is

18 + 21 + 14 + 31 = 84.