Perimeter of an equilateral triangle is 4 inches more than the perimeter of a​ square, and the side of the triangle is 4 inches longer than the side of the square

Question

The perimeter of an equilateral triangle is 4 inches more than the perimeter of a​ square, and the side of the triangle is 4 inches longer than the side of the square. Find the side of the triangle.​ (Hint: An equilateral triangle has three sides the same​ length.)

Answer 1

side of the triangle = 12 inches

side of the square = 8 inches

12 is 4 more than 8

perimeter of the square = 4(8) =32 inches

perimeter of the triangle = 3(12) = 36 inches

36 is 4 more than 32

Let T= side of the triangle and S = side of the square

3T = 4S+4

T=S+4

3(S+4) = 4S+4

3S+12 = 4S +4

4S-3S = S = 12-4 = 8

T =8+4 = 12

Answer 2

I'm going to use Ps = Perimeter of the square and Pt = Perimeter of the triangle

We are given that Pt = Ps + 4

Say t = triangle side and s = square side

We're given that t = s + 4

Since we're dealing with an equilateral triangle, we know that all the sides have the same length, so their perimeter can be written as Pt = 3t

Similarly, we can write the perimeter of the square as Ps = 4s

Let's do some substitution with what we have so far:

Pt = 3t = 3(s+4)

Our 1st equation was Pt = Ps + 4

So Ps + 4 = 3(s+4)

We also derived that Ps = 4s

So 4s + 4 = 3(s+4)

Solve for s:

4s + 4 = 3s + 12

s = 8

Using that, we know that t = s + 4, so t = 12

Checking that against the perimeters, Pt = 36 and Ps = 32. They're 4 apart, so it check out.

Answer 3

Approach: Translate. This kind of problem can get a little messy if you don't stay organized, so start writing down all those words into math. Here's how I did it:

The perimeter of an equilateral triangle is 4 inches more than the perimeter of a square,...

Okay. Stop there. Triangle who? Square where? Don't worry so much about the shapes, think about what you already know about the shapes. Equilateral triangle? Has sides of equal length (that is in the hint later, but we haven't technically read that far yet). Remember how it also mentions perimeter (the sum of the sides of a shape)? Well that gives us our first equation. Let's call A the perimeter of the triangle, and L the length of one of the triangle's legs. Then

A=3L.

Great! So it also mentions a square and its perimeter. Actually, this follows a strikingly similar train of reasoning, except that a 4 equal sides instead of 3. Let's call B the perimeter of the square, and S one side length of the square. Equation?

B=4S.

Excited? I am too. But don't get so excited that you move on quite yet--there's still one more equation we can pull from this. Re-read the snippet, and you'll notice that there's a bigger picture here: a relation between the shapes. Notice how we've already assigned variables to every item in that phrase, so we can translate it right into an equation:

A=B+4.

Great stuff! Moving on.

...and the side of the triangle is 4 inches longer than the side of the square.

Oh, these are all familiar terms, aren't they? We even gave each of them a letter already!

L=S+4.

Find the side of the triangle.

What are we looking for? That's right. We want to solve for L. What do we know?

A=3L

B=4S

A=B+4

L=S+4

We just need to play around with the equations we've written: dancing around the equals sign. Why not start with replacing B with 4S in the third equation. Based on our second equation, we know that's an equivalent swap!

A=4S+4

Hmm, don't we also have another equation with A? Yes! For sure! We know that A=3L. So why not replace this in the equation above to get:

3L=4S+4.

The last thing we know is that L=S+4. Sure, we could substitute S+4 into that new equation in place of L, except we're solving for L. Eliminating that variable isn't going to help us solve it as quickly. (We still can, but it's not as direct. Feel free to ask about how that would work if you're curious!) Instead, let's rearrange this equation to state that

S=L-4.

Replace this in the equation with L to get:

3L=4(L-4)+4.

Almost there! Now for a few arithmetic operations and we'll be all set.

3L=4L-16+4

3L=4L-12

L=12

That's it! 12 what? The side of the triangle is 12 inches. Remember that the question is asking about lengths here, so be sure to label your units.