1986 AMC 12 Problem 26

Question

It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the

x

and

y

axes and so that the medians to the midpoints of the legs lie on the lines

y=3x+1

and

y=mx+2.

The number of different constants

m

for which such a triangle exists is

0

1

2

3

more than

3

Answer: C

Concepts:coordinate geometry median slope construction

Difficulty rating: 2230

Small hint:

Place an axis-aligned right triangle at convenient coordinates and compute the slopes of the two medians to its legs

Big hint:

The two slopes differ by a factor of

4;

remember that either given line could be the steeper one

Solution:

Place the right-angle vertex at

(0,0)

and the other vertices at

(a,0)

and

(0,b).

The medians to the legs have slopes

-\frac{2b}{a}

and

-\frac{b}{2a},

whose ratio is

4.

Therefore, if one median has slope

3,

the other can have slope

12

or

\frac{3}{4}.

Both occur: choose a triangle with the required pair of slopes and translate its centroid to the intersection of the two specified lines. Thus there are two possible values of

m.

Therefore the correct answer is C.