This Triangle Shouldn't Exist - But It Does
A Right-Angled Triangle That Breaks the Rules
For a moment, I wondered: how possible is it to find the side lengths of a right-angled triangle when all we know are its perimeter and area?
It’s easy enough with a rectangle or a square — but a triangle? That’s a whole different story. Let’s see what happens.
To crack this puzzle, we’ll set up a system of equations with three variables representing the triangle’s sides. Then, with some clever algebraic tricks, we’ll find their values one by one.
Maybe you’ve got a better approach — I’d love to see it!
Ready? Let’s jump right in.
How do we set up this system of equations to help us find the side lengths of this triangle?
To begin, let’s label the sides of the triangle
In any shape, the perimeter is simply the sum of the lengths of its sides. This means that for the shape, the perimeter is:
a + b + c = 45
Now let’s consider the area. For a triangle, the formula is:
1/2 x base x height. So in this case, the area becomes:
1/2 x b x a = 45
One more thing!
Because it’s a right-angled triangle, we can apply the Pythagorean theorem:
a² + b² = c²
At this point, we’ve got three equations and three unknowns — one for each side of the triangle.
Amazing! 👏👏👏
Now comes the tricky part — solving these equations together to find the side lengths.
Let us begin with the first equation. It can also be written as:
a + b = 45 — c
When we square both sides, we get this:
(a + b)² = (45 — c)²
From here, we’ll work with these identities:
So, on expansion, we get:
But remember a² + b² = c², and ab = 90. So, we can make the substitutions right away!
Subtracting c² from both sides, we’re left with:
180 = 2025 — 90c
Simplifying further, we get c = 20.5
Cool!
Now, we know the value of one of the unknowns. Let us substitute this value in the first equation.
This means a = 24.5 — b.
We can replace this value in the third equation: a² + b² = c²
On expansion, we get this:
From here, we’ll use the quadratic formula to solve for b, and it gives us two possible values: 20 and 20.5.
Cool!
We’re very close to the finish line!
Since we have two possible values for b, we’ll also get two corresponding values for a.
To find them, let’s use the second equation: 1/2ab = 45.
When we make the substitutions, we get this:
For b = 20, a = 4.5; and for b = 4.5, a = 20
So, we end up with two possible sets of side lengths:
a = 4.5, b = 20, and c = 20.5
a = 20, b = 4.5, and c = 20.5
👏👏👏👏
We can verify our answer using the Pythagorean theorem: the square of the hypotenuse should equal the sum of the squares of the other two sides.
(20.5)² = (4.5)² + 20²
420.25 = 20.25 + 400
420.25 = 420.25 ✔✔✔
Thanks for reading! If you liked this puzzle, please give a few claps, follow me, and don’t forget to subscribe to stay updated with my latest articles.
You can also follow me on X (formerly Twitter) for more exciting challenges.
Feel free to reach out to me here. Let’s connect!