1958 AMC 12 Problem 19

Attempt Problem 19 of the 1958 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1958 AMC 12 solutions, or check the answer key.

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19.

The sides of a right triangle are

a

and

b

and the hypotenuse is

c.

A perpendicular from the vertex divides

c

into segments

r

and

s,

adjacent respectively to

a

and

b.

a:b=1:3,

then the ratio of

r

s

is:

1:3

1:9

1:10

3:10

1:\sqrt{10}

Use the projection relations

a^2=cr

and

b^2=cs

Divide the two projection equations to express

\frac{r}{s}

in terms of

\frac{a}{b}

Similarity from the altitude-to-hypotenuse construction gives

a^2=cr,\qquad b^2=cs.

Therefore

\frac rs=\frac{a^2}{b^2} =\left(\frac13\right)^2=\frac19.

Thus, the correct answer is B.

Problem 19 in other years