Triangle Sides Calculation: Perimeter & Side Sum
Hey guys! Let's dive into a geometry problem where we need to figure out the sides of a triangle. We're given some clues: we know the perimeter, the sum of two sides, and one side's length. This is a classic math puzzle, so let's break it down step-by-step. We will learn how to find the missing sides of a triangle using the perimeter and the sum of two sides. We'll be using the information provided to unravel the mystery of the triangle's dimensions. It's like being a geometry detective, figuring out the hidden lengths! This is a super common type of problem, and understanding it will boost your overall math skills. This problem involves some basic algebra and geometric principles. So, grab your pencils and let's get started.
We will be discussing how to find the sides of a triangle given its perimeter and some information about the sides. Specifically, we're dealing with a triangle ABC, and here's what we know:
- Side BC = 14.5 cm
- AB + AC = 21.75 cm
- Perimeter of triangle ABC = 29 cm
Our mission is to find the lengths of the sides AB and AC. Remember, the perimeter of a triangle is simply the sum of the lengths of all its sides. This means: Perimeter = AB + BC + AC. With the given values, we're on a mission to crack this code and learn a valuable mathematical concept.
Understanding the Basics: Perimeter and Sides
Alright, before we get our hands dirty with the calculations, let's make sure we're all on the same page. The perimeter of any shape is just the total distance around its outside. For a triangle, that means adding up the lengths of all three sides. In our case, the triangle has sides AB, BC, and AC. We know the total distance around the triangle (the perimeter) is 29 cm. We also have some clues about the sides: we know one side (BC) is 14.5 cm, and the sum of the other two sides (AB + AC) is 21.75 cm. So, let’s use this information to our advantage.
Now, let's look at how to approach this problem systematically. To solve this, we can use the formula for the perimeter of a triangle, which is: Perimeter = AB + BC + AC. We're given the perimeter, which is 29 cm. We also know that BC = 14.5 cm. And we know that AB + AC = 21.75 cm. Now, we want to solve for AB and AC, which is basically the meat of the problem. This is where we need to use a little bit of algebra to rearrange the formula to find the missing lengths. It is pretty simple and you will get the hang of it easily. Let's see how we can put it all together. This will help you in your math class and even when you go on to higher studies. Ready?
So, as we've said, the perimeter (P) of a triangle is the sum of its sides. In our case, P = AB + BC + AC. We're given the values: P = 29 cm, BC = 14.5 cm, and AB + AC = 21.75 cm. Now, let’s go ahead and find the length of the sides.
Finding the Lengths of AB and AC: The Calculations
Alright, let's put on our math hats and solve this triangle mystery! We've got all the pieces of the puzzle; now, it's time to fit them together. We know that the perimeter of the triangle is 29 cm, and this perimeter is equal to AB + BC + AC. Since we know BC, and the sum of AB + AC, we can use those facts to solve for what we are looking for. Because we know that AB + AC is equal to 21.75 cm. We can substitute that value into our perimeter equation. The perimeter of the triangle (P) is 29 cm. We know BC = 14.5 cm, and AB + AC = 21.75 cm. Let's make it work:
- Start with the Perimeter Formula: P = AB + BC + AC
- Substitute the known values: 29 cm = AB + 14.5 cm + AC
- Combine the known information: We know AB + AC = 21.75 cm. We can replace (AB + AC) in the perimeter equation.
- Rewrite the equation: 29 cm = 21.75 cm + BC
- Solve for BC: BC = 29 cm - 21.75 cm. Therefore, BC = 7.25 cm.
Now that we have the length of BC (7.25 cm), we can double-check our work using the original equation. Let’s do it step by step so it's easy to follow:
- We know that the perimeter is 29 cm.
- We know BC is 14.5 cm, and we just found out that AC is 7.25 cm.
- Substituting these values into the original perimeter equation: AB + BC + AC = 29 cm.
- AB + 14.5 cm + 7.25 cm = 29 cm.
- Solving for AB: AB = 29 cm - 14.5 cm - 7.25 cm.
- Therefore, AB = 7.25 cm.
So, the sides are: AB = 7.25 cm, AC = 14.5 cm, and BC = 7.25 cm. It's awesome when everything works out! Keep practicing, and these kinds of problems will become second nature to you. You've got this! Also, you'll be able to quickly solve these types of questions with confidence. Remember, the key is to understand the formulas and practice consistently. Now, let us check our answer. Since we know that AB + AC = 21.75 cm, and now we know that BC = 7.25 cm, then the value of AC is equal to 21.75cm - 7.25 cm = 14.5 cm. This is exactly what we got.
Verification and Conclusion
Great job, guys! We've found the lengths of all three sides of the triangle! Let's just do a quick recap and make sure everything adds up. We have found out that AB = 7.25 cm, AC = 14.5 cm and BC = 7.25 cm. The perimeter is the total length around the triangle, so if we add up all the sides, we should get the perimeter of 29 cm that we were given. Let's add them up: AB + BC + AC = 7.25 cm + 14.5 cm + 7.25 cm = 29 cm. This matches the perimeter given in the problem, so our answer is correct!
In Conclusion:
We successfully found the lengths of sides AB and AC using the given information: the perimeter of the triangle, the length of side BC, and the sum of sides AB and AC. This is a great example of how you can use basic math principles to solve a geometry problem. Keep practicing these types of problems, and you'll become a geometry whiz in no time. Always remember to break down the problem, use the formulas, and double-check your work. You are well on your way to mastering geometry! The key takeaways from this exercise are understanding how to utilize the perimeter and the sum of sides to find the individual lengths of the sides of a triangle. Understanding these core concepts is essential for a strong foundation in geometry. So, keep up the amazing work! You are now equipped with the knowledge and skills to tackle similar problems with confidence. Keep practicing and exploring different types of geometry problems, and soon you'll be solving complex questions easily! Remember, the more you practice, the better you get. You've got this! Also, don't hesitate to ask your teacher or classmates for help if you are stuck.