1962 AMC 12 Problem 23

Question

In triangle

ABC,

CD

is the altitude to

AB

and

AE

is the altitude to

BC.

If the lengths of

AB,

CD,

and

AE

are known, the length of

DB

is:

not determined by the information given

determined only if

A

is an acute angle

determined only if

B

is an acute angle

determined only if

ABC

is an acute triangle

none of these is correct

Compute the triangle’s area in two ways using the two known altitudes

After finding

BC,

use right triangle

BCD

Let

AB=c,

CD=h,

and

AE=e.

Equating two area formulas gives

\frac12ch=\frac12(BC)e,

so

BC=\frac{ch}{e}.

Since triangle

BCD

is right at

D,

DB=\sqrt{BC^2-CD^2}.

This determines

DB

whether the original triangle is acute, right, or obtuse, so none of the first four choices is correct.

Thus, the correct answer is E.

Problem 23 in other years