The triangle below lies on a flat surface and is pushed at the top vertex. The length of the congruent sides
does not change, but the angle between the two congruent sides θ will increase, and the base will stretch.
Initially, the area of the triangle will increase, but eventually the area will decrease, continuing until the
triangle collapses.
What is the maximum area achieved during this process? And, what is the length of the base when this
process is used to create a different triangle whose area is the same as the triangle above?
20
20
θ
24
Solution:
Consider that the triangle is rotated counterclockwise so that it one of the congruent sides is the
base as show in the figure below. Consider the triangle ABC. Let h be the height of the triangle.
h
24
20
A
θ
C
B
20
We know that sinθ = AB/AC = h/20 or h = 20 x sinθ.
We also know that Area of the triangle = 1/2 x b x h, where b = base of the triangle and h = height of the
triangle.
In the triangle base b = 20 units and height h = 20 x sinθ. So we can area of the triangle
A = 1/2 x 20 x 20 x sinθ.
Question 1:
We are supposed to find the max area of the triangle. The max area in the above equation is
possible when sinθ = 1.
Sinθ = 1 when θ, the angle between the congruent sides = 900.
So maximum value of the area Amax = ½ x 20 x 20 = 200 sq units
Question 2:
Consider the triangle below. Draw a line AD perpendicular to the base BC. Since AB = AC = 20 units, the
line AD will bisect the base BC. So BD =DC=12 units. Consider the triangle ABD. Using the Pythagoras
theorem AB2 = BD2+AD2 or AD2= AB2-BD2 or AD2 = 202-122 or AD2 = 156 or AD =16 units = height of the
triangle
We can write the area of the triangle = ½ b x h where b = 24 units and h = 16 units. So A = ½ x 24 x 16=
192 sq units.
A
θ
20
20
12
B
900
D
24
C
We are supposed to find the base of triangle when the area of the triangle will be the same as the
current area of the triangle.
This can happen when h = 12 units and base = 32 units
Area = ½ x 32 x 12 = 192 sq units
Below is the graph of the area of the triangle at different base lengths
Below is the graph of the area of the triangle at different inclusive angles between the sides.