Question

The congruent sides of an isosceles triangle are each 1 unit longer than the length of the shortest side of the triangle. The perimeter of the triangle is the same as the perimeter of a square whose side length is 2 units shorter than the length of the shortest side of the triangle. What is the length of the shortest side of the triangle?

___=units

Answer 1

Let's call the long sides of the triangle A and the short sides of the triangle B. We know that the perimeter of the triangle is 2A+B. We also know that A=B+1 (long sides are 1 unit longer than the short side).
Now, the confusingly-worded portion "The perimeter of the triangle is the same as the perimeter of a square whose side length is 2 units shorter than the length of the shortest side of the triangle."
This just means we can set 2A+B equal to the perimeter of this square. The square has sides equal to "2 units shorter than the shortest side", so each side of the square is B-2. A square has 4 sides, so the perimeter would be 4(B-2) or 4B-8.
So, we get 2A+B=4B-8. Next, how about we substitute A for B+1?
2(B+1)+B=4B-8
Can you simplify this equation to find the value of B? That should give you the length of the short side of the triangle.

Answer 2

uhhhhh...

Answer 3

im just even more confused now

Answer 4

we didnt learn anything to do with shapes until this question so thats probably why i dont undersdtand a thing

Answer 5

Yeah, it was difficult for me to visualize as a shape, so that's why I tried to set it up like an algebra problem. Have you learned much of algebra already?

Answer 6

well im in algebra 1A

Answer 7

I tried to draw out my explanation. I realize it's tricky to keep track of all these variables without some reference.

Answer 8

We have the two long sides of the triangle, and they have some length we don't know. But we *do* know that the long sides are both longer than the short side by 1. So, if we take the long side, and subtract 1, we get the length of the short side. In the same way, if we take the short side and add 1, we get the length of the long side.

Answer 9

thats alot of understanding just to type in one number

Answer 10

Next, there's this square that the problem mentions. There's 4 sides to a square, so we know the perimeter of a square is the same as 4 times the length of one side.
Now, we don't know the length of the square's edges either, but we *do* know that they are shorter than the triangle's *short* side by 2.

Answer 11

I agree, this problem sure doesn't slack when it comes to handiwork. You're probably wondering if there's an easier way to go about this. There sure is, but I'm being a little over-explanatory for the sake of understanding. But I'll show you how I'd go about solving the problem if it were just myself.

Answer 12

thats sounds better

Answer 13

Our final goal is the *short side of the triangle*. Let's call this X. And this time, I'll only use the variable X, to keep things simple.

Answer 14

mk

Answer 15

ugh i hate this class

Answer 16

The perimeter of these two shapes are equal, so let's set up equations to represent them.
First the triangle. The perimeter of the triangle is X+(X+1)+(X+1). This represents adding the short side (X) with the two long sides (X+1). This simplifies to 3X+2.
Then we have the square. The perimeter of the square is 4*(X-2) because each side of the square is "2 units shorter than the short end of the triangle". Simplify this with distribution to get 4X-8.
Since the perimeters are equal, we can say 3X+2=4X-8. Get all the X's on one side and all the numbers on the other. You can do this by first subtracting 3X from both sides.
This gives you 3X-3X+2=4X-3X-8, which simplifies to 2=X-8. Then, add 8 to both sides. You get 2+8=X-8+8, which simplifies to 10=X. This is the same as X=10, so there we have it. The length of the short side would be 10.

Answer 17

You can double-check this by calculating the perimeters of the triangle and square. If the short side is 10, then the long sides of the triangle would be 11, and the sides of the triangle would be 8. The perimeter of the triangle would be 10+11+11=32; the perimeter of the square would be 8+8+8+8=32. The perimeters are equal after all, like the problem said they should be. So we know we did the problem correctly.

Answer 18

ill guess

Answer 19

You really don't have to! I've worked it out for you!
Please let me know if there's something I can help you understand. I'll try to explain better if I need to.