In triangle ABC, ∠ C=90°, AC=6 and BC=8. Points D and E are on AB and BC, respectively, and ∠ BED=90°. If DE=4, then BD=

1995 AMC 12 Problem 8

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In triangle

ABC,

\angle C=90^\circ,

AC=6

and

BC=8.

Points

D

and

E

are on

\overline{AB}

and

\overline{BC},

respectively, and

\angle BED=90^\circ.

DE=4,

then

BD=

5

\frac{16}3

\frac{20}3

\frac{15}2

8

The two right angles show that

\overline{DE}\parallel\overline{AC}

Use similarity between

\triangle BDE

and

\triangle BAC

The large right triangle has

AB=\sqrt{6^2+8^2}=10.

Since

\overline{DE}\parallel\overline{AC},

triangles

BDE

and

BAC

are similar, with scale factor

\frac{DE}{AC}=\frac{4}{6}=\frac{2}{3}.

Therefore

BD=(\frac{2}{3})(10)=\frac{20}{3}.

Thus the correct answer is C.