# 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

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

This Demonstration shows a construction of a triangle given the sum of the lengths of the sides and , the angle at and the length of the altitude from to the base .

ABC

s

CA

BC

α

A

h

C

C

AB

Construction

Step 1: Draw a horizontal ray from and a ray from at angle from . On , measure out a point at distance from and a point at distance from .

ρ

1

A

ρ

2

A

α

ρ

1

ρ

2

C

h

C

ρ

1

D

s

A

Step 2: Let and be the intersections of the circle with center and radius .

B'

B''

C

CD

Then either of the two triangles or is a solution.

AB'C

AB''C

Verification

Consider the triangle . Since , the first condition is satisfied.

AB'C

s=AD=AC+CD=AC+B'C

Since the distance of to is by construction, the length of the altitude from to the base is .

C

AB'

h

C

C

AB'

h

C