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