Solve This Triangle Perimeter Puzzle!

by Andrew McMorgan 38 views

Hey guys! Today, we're diving into a classic math problem that's all about triangles and their perimeters. You know, those shapes that are everywhere, from pizza slices to road signs? We've got a triangle here with a perimeter of 55 cm, and we need to figure out the lengths of its shortest, middle, and longest sides. It sounds a bit like a detective story, right? We've got clues, and we need to piece them together to find our missing lengths. This problem is perfect for anyone looking to sharpen their algebra skills or just flex those brain muscles. So, grab your notebooks, and let's break down this triangle perimeter problem together. We'll tackle it step-by-step, making sure we understand each part of the puzzle. Remember, the perimeter of any shape is simply the total distance around its outer edge. For a triangle, this means adding up the lengths of all three sides. Our total is 55 cm, which gives us a solid starting point. The real challenge comes from the relationships between the sides. We're told the shortest side is related to the middle side, and the longest side is related to the other two. This is where algebra comes in handy. We'll use variables to represent the unknown lengths and set up equations based on the information given. Don't worry if algebra isn't your favorite subject; we'll keep it simple and clear. By the end, you'll not only have the answer but also a better grasp of how to solve similar triangle perimeter problems. Let's get started on unraveling this geometric mystery!

Understanding the Clues: Setting Up Our Triangle

Alright, let's get down to business with our triangle perimeter problem. We know the total perimeter is 55 cm. That's our big number, the grand total we're working towards. Now, let's look at the relationships between the sides. We've got a shortest side, a middle side, and a longest side. Let's give them some names, or rather, some algebraic variables. It's usually easiest to start by defining the smallest or simplest unknown. Let's say the length of the middle side is 'm' cm. Why the middle side? Sometimes it's easier to define things relative to a central value, and in this case, it helps us relate both the shortest and longest sides. So, we have:

  • Middle Side = m cm

Now, the first clue tells us: "The measure of the shortest side is 8 cm less than the middle side." If the middle side is 'm', then the shortest side must be 'm - 8' cm. Easy, right? So now we have:

  • Shortest Side = (m - 8) cm

This makes sense because the shortest side must be smaller than the middle side, and subtracting 8 certainly achieves that. Keep in mind, for this to be a valid triangle side, 'm' must be greater than 8, otherwise, the shortest side would have a zero or negative length, which is impossible in geometry. We'll keep this in mind as we go.

Next, we look at the longest side. The clue here is: "The measure of the longest side is 1 cm less than the sum of the other two sides." The other two sides are our shortest side (m - 8) and our middle side (m). So, their sum is (m - 8) + m. Simplifying that gives us 2m - 8. The longest side is 1 cm less than this sum. Therefore:

  • Longest Side = (2m - 8) - 1 cm

Simplifying the longest side expression, we get:

  • Longest Side = (2m - 9) cm

So, to recap, we've defined all three sides in terms of 'm':

  • Shortest Side: (m - 8) cm

  • Middle Side: m cm

  • Longest Side: (2m - 9) cm

This setup is crucial for solving our triangle perimeter problem. We've translated the word problem into algebraic expressions. The next step, which is often the most powerful in these types of problems, is to use the perimeter information to create a single equation. Remember, the perimeter is the sum of all the sides. We'll plug these expressions into the perimeter formula, and that's where the magic happens!

Building the Equation: The Power of the Perimeter

Alright, guys, we've successfully broken down the clues and defined our triangle's sides using algebra. Now comes the part where we bring it all together and actually solve for those lengths. This is where the triangle perimeter problem really starts to take shape, pun intended! We know that the perimeter of a triangle is the sum of the lengths of its three sides. We are given that the total perimeter is 55 cm. So, we can set up an equation by adding our expressions for the shortest, middle, and longest sides and setting that sum equal to 55.

Let's write it out clearly:

(Shortest Side) + (Middle Side) + (Longest Side) = Perimeter

Substituting our expressions:

(m - 8) + (m) + (2m - 9) = 55

See how we've got everything in terms of 'm' now? This is the beauty of algebra! It allows us to take a complex situation with multiple unknowns and boil it down to a single equation with a single variable. Our goal now is to solve this equation for 'm'.

First, let's combine the like terms on the left side of the equation. We have 'm' terms and constant terms (numbers).

  • Combine the 'm' terms: m + m + 2m = 4m
  • Combine the constant terms: -8 - 9 = -17

So, the equation simplifies to:

4m - 17 = 55

Now, we want to isolate 'm'. To do this, we'll use inverse operations. First, we need to get rid of the '-17' on the left side. The opposite of subtracting 17 is adding 17. So, we'll add 17 to both sides of the equation to keep it balanced:

4m - 17 + 17 = 55 + 17

This simplifies to:

4m = 72

Almost there! Now, 'm' is being multiplied by 4. To isolate 'm', we do the opposite of multiplying by 4, which is dividing by 4. Again, we must do this to both sides:

4m / 4 = 72 / 4

This gives us:

m = 18

Boom! We've found the value of 'm'. Remember, 'm' represents the length of the middle side in centimeters. So, the middle side is 18 cm. This is a huge step in solving our triangle perimeter problem! We've used the perimeter and the relationships between the sides to find one of the key measurements. Now that we know 'm', we can easily find the lengths of the shortest and longest sides. Let's move on to the final calculation and confirm our answer!

Finding the Sides and Confirming the Solution

Alright, legends, we've done the heavy lifting! We've used the clues and the power of algebra to find that m = 18. This means our middle side is 18 cm. Now, let's use this value to find the lengths of the other two sides: the shortest and the longest. This is the final nail in the coffin for our triangle perimeter problem!

Remember how we defined the sides earlier?

  • Middle Side = m cm

So, the Middle Side is 18 cm.

  • Shortest Side = (m - 8) cm

Substitute m = 18:

Shortest Side = (18 - 8) cm

Shortest Side = 10 cm

Looking good! The shortest side is indeed smaller than the middle side.

  • Longest Side = (2m - 9) cm

Substitute m = 18:

Longest Side = (2 * 18 - 9) cm

Longest Side = (36 - 9) cm

Longest Side = 27 cm

So, we have our three side lengths: 10 cm, 18 cm, and 27 cm. But hold up! Before we celebrate, we need to do one crucial thing: confirm our answer. Does this solution actually satisfy all the conditions of the problem? Let's check!

  1. Is the perimeter 55 cm? Add the lengths: 10 cm + 18 cm + 27 cm = 55 cm. YES! The perimeter matches.

  2. Is the shortest side 8 cm less than the middle side? Middle side is 18 cm, shortest is 10 cm. 18 - 8 = 10. YES! This condition is met.

  3. Is the longest side 1 cm less than the sum of the other two sides? The other two sides are 10 cm and 18 cm. Their sum is 10 + 18 = 28 cm. The longest side is 27 cm. Is 27 cm equal to 1 cm less than 28 cm? Yes, 28 - 1 = 27. YES! This condition is also met.

All conditions are satisfied! This means our calculations are correct, and the lengths of the sides of the triangle are 10 cm, 18 cm, and 27 cm. Pretty awesome, right? You just conquered a triangle perimeter problem using algebra. Keep practicing these, and you'll be a geometry whiz in no time!