14. Construct a Triangle Given the Length of the Altitude to the Base, a Base Angle and the Sum of the Lengths of the Other Two Sides

s

α

h

C

step 1

step 2

This Demonstration shows a construction of a triangle

ABC

given the sum

s

of the lengths of the sides

CA

and

BC

, the angle

α

at

A

and the length

h

C

of the altitude from

C

to the base

AB

.

Construction

Step 1: Draw a horizontal ray

ρ

1

from

A

and a ray

ρ

2

from

A

at angle

α

from

ρ

1

. On

ρ

2

, measure out a point

C

at distance

h

C

from

ρ

1

and a point

D

at distance

s

from

A

.

Step 2: Let

B'

and

B''

be the intersections of the circle with center

C

and radius

CD

.

Then either of the two triangles

AB'C

or

AB''C

is a solution.

Verification

Consider the triangle

AB'C

. Since

s=AD=AC+CD=AC+B'C

, the first condition is satisfied.

Since the distance of

C

to

AB'

is

h

C

by construction, the length of the altitude from

C

to the base

AB'

is

h

C

.