Problem of the Week

Problem E and Solution

Terry’s Triangles

Problem

Terry is drawing isosceles triangles with side lengths $$a$$, $$b$$, and $$c$$ such that \begin{aligned} a&=y-x\\ b&=x+z\\ c&=y-z\end{aligned} where $$x$$, $$y$$, and $$z$$ are positive integers and $$x+y+z<10$$.
Find all the possible triples $$(a,b,c)$$ that satisfy this.

Solution

In an isosceles triangle, two sides must have equal length. So we need to consider three cases: $$a=b$$, $$b=c$$, and $$a=c$$. Also, in order for $$a$$, $$b$$, and $$c$$ to represent side lengths of a triangle, they must be positive numbers and the sum of any two side lengths must be greater than the other side length.
Case 1: $$a=b$$
If $$a=b$$, then $$y-x=x+z$$, so $$y=2x+z$$. We can make a table of all the values of $$x$$, $$y$$, and $$z$$ that satisfy this equation as well as $$x+y+z<10$$, and then find the corresponding values of $$a$$, $$b$$, and $$c$$ and check if they are valid side lengths.

$$x$$ $$y$$ $$z$$ $$a$$ $$b$$ $$c$$ Valid?
1 3 1 2 2 2 Yes
1 4 2 3 3 2 Yes
1 5 3 4 4 2 Yes
2 5 1 3 3 4 Yes

Case 2: $$b=c$$
If $$b=c$$, then $$x+z=y-z$$, so $$y=x+2z$$. As in Case 1, we can write the possible values of $$x$$, $$y$$, $$z$$, $$a$$, $$b$$, and $$c$$ in a table.

$$x$$ $$y$$ $$z$$ $$a$$ $$b$$ $$c$$ Valid?
1 3 1 2 2 2 Yes
2 4 1 2 3 3 Yes
3 5 1 2 4 4 Yes
1 5 2 4 3 3 Yes

Case 3: $$a=c$$
If $$a=c$$, then $$y-x=y-z$$, so $$x=z$$. As in previous cases, we can write the possible values of $$x$$, $$y$$, $$z$$, $$a$$, $$b$$, and $$c$$ in a table.

$$x$$ $$y$$ $$z$$ $$a$$ $$b$$ $$c$$ Valid?
1 1 1 0 2 0 No ($$a$$ and $$c$$ are not positive)
1 2 1 1 2 1 No ($$a+c \not>b$$)
1 3 1 2 2 2 Yes
1 4 1 3 2 3 Yes
1 5 1 4 2 4 Yes
1 6 1 5 2 5 Yes
1 7 1 6 2 6 Yes
2 1 2 $$-1$$ 4 $$-1$$ No ($$a$$ and $$c$$ are not positive)
2 2 2 0 4 0 No ($$a$$ and $$c$$ are not positive)
2 3 2 1 4 1 No ($$a+c \not>b$$)
2 4 2 2 4 2 No ($$a+c \not>b$$)
2 5 2 3 4 3 Yes
3 1 3 $$-2$$ 6 $$-2$$ No ($$a$$ and $$c$$ are not positive)
3 2 3 $$-1$$ 6 $$-1$$ No ($$a$$ and $$c$$ are not positive)
3 3 3 0 6 0 No ($$a$$ and $$c$$ are not positive)
4 1 4 $$-3$$ 8 $$-3$$ No ($$a$$ and $$c$$ are not positive)

Therefore, there are 12 possible triples $$(a,b,c)$$. They are listed below. \begin{aligned} &(2,2,2)\quad(3,3,2)\quad(4,4,2)\quad(3,3,4)\\ &(2,3,3)\quad(2,4,4)\quad(4,3,3)\\ &(3,2,3)\quad(4,2,4)\quad(5,2,5)\quad(6,2,6)\quad(3,4,3)\end{aligned}