Problem of the Week

Problem D and Solution

Different Lengths

Problem

$$\triangle ABC$$ is isosceles with $$AB=AC$$. All three side lengths of $$\triangle ABC$$ and also altitude $$AD$$ are positive integers.
If the area of $$\triangle ABC$$ is 60 cm$$^2$$, determine all possible perimeters of $$\triangle ABC$$.

Note: You may use the fact that the altitude of an isosceles triangle drawn to the unequal side bisects the unequal side.

Solution

Let the base of $$\triangle ABC$$ (which is $$BC$$) have length $$b$$ and the equal sides ($$AB$$ and $$AC$$) have length $$c$$, as shown in the diagram.

The area of $$\triangle ABC$$ is $$\dfrac{\text{base} \times \text{height}}{2}=\dfrac{bh}{2}$$.
Since this area is given to be 60 cm$$^2$$, we have $$\dfrac{bh}{2}=60$$ or $$bh = 120$$.
We are given that $$b$$ and $$h$$ are positive integers. We will consider the positive factors of 120 to generate all possibilities for $$b$$ and $$h$$. Since the altitude $$AD$$ bisects $$BC$$, $$\triangle ABC$$ is composed of two congruent right-angled triangles, each with side lengths $$c, h$$, and $$\frac{b}{2}$$. We will use the Pythagorean Theorem in one of these right-angled triangles to generate a value of $$c$$ for each possibility.

$$h$$ $$b$$ $$\frac{b}{2}$$ $$c^2=h^2+(\frac{b}{2})^2$$ Valid?
1 120 60 3601 No, $$c$$ is not an integer
2 60 30 904 No, $$c$$ is not an integer
3 40 20 409 No, $$c$$ is not an integer
4 30 15 241 No, $$c$$ is not an integer
5 24 12 169 Yes, $$c=13$$
6 20 10 136 No, $$c$$ is not an integer
8 15 7.5 120.25 No, $$c$$ is not an integer
10 12 6 136 No, $$c$$ is not an integer
12 10 5 169 Yes, $$c=13$$
15 8 4 241 No, $$c$$ is not an integer
20 6 3 409 No, $$c$$ is not an integer
24 5 2.5 582.25 No, $$c$$ is not an integer
30 4 2 904 No, $$c$$ is not an integer
40 3 1.5 1602.25 No, $$c$$ is not an integer
60 2 1 3601 No, $$c$$ is not an integer
120 1 0.5 14 400.25 No, $$c$$ is not an integer

We see that there are two solutions for $$(h,b,c)$$. They are $$(5,24,13)$$ and $$(12,10,13)$$.
The side lengths of the corresponding triangles are 24, 13, and 13 and 10, 13, and 13. Therefore, the perimeter of $$\triangle ABC$$ is either 50 cm or 36 cm.