15. Construct a Triangle Given the Lengths of Two Sides and the Median to the Third Side

a

b

m

C

step 1

step 2

This Demonstration shows a construction of a triangle

ABC

given the lengths

a

and

b

of the sides

BC

and

CA

and the length of the median

m

C

from

C

to

AB

.

Construction

Let

EC

be a line segment of length

2

m

C

and midpoint

D

.

Step 1: Draw a circle with center

E

and radius

b

and a circle with center

C

and radius

a

. Let

B

and

B'

be the intersections of the two circles.

Step 2: Extend

BD

to the point

A

such that

BD=DA

.

Verification

The triangle

ABC

has side

BC

of length

a

.

EC

and

BA

bisect each other, so

EACB

is a parallelogram with sides

a

and

b

. The side

CA

has length

b

, and the line segment

DC

is the median from

C

of length

m

C

.