The Shortest Side Of A Triangle Is $s$ Centimeters Long. The Longest Side Is 8 Cm Longer Than The Shortest Side. The Third Side Is 12 Cm Long. The Perimeter Is 38 Cm. Find The Length Of The Shortest Side.Which Equation Represents The

Introduction

In this article, we will delve into a mathematical puzzle involving a triangle with three sides of different lengths. The shortest side is given as $s$ centimeters, while the longest side is 8 cm longer than the shortest side. The third side is 12 cm long, and the perimeter of the triangle is 38 cm. Our goal is to find the length of the shortest side, denoted by $s$. We will use algebraic equations to represent the relationships between the sides of the triangle and solve for the value of $s$.

Understanding the Problem

Let's break down the information given in the problem:

• The shortest side of the triangle is $s$ centimeters long.
• The longest side is 8 cm longer than the shortest side, so its length is $s + 8$ cm.
• The third side is 12 cm long.
• The perimeter of the triangle is 38 cm, which means the sum of all three sides is 38 cm.

We can represent the perimeter of the triangle using the following equation:

$$s + (s + 8) + 12 = 38$$

Simplifying the Equation

To solve for $s$, we need to simplify the equation by combining like terms. Let's start by adding the constants on the left-hand side of the equation:

$$s + s + 8 + 12 = 38$$

This simplifies to:

$$2s + 20 = 38$$

Next, we can subtract 20 from both sides of the equation to isolate the term with $s$:

$$2s = 18$$

Solving for $s$

Now that we have isolated the term with $s$, we can solve for its value by dividing both sides of the equation by 2:

$$s = \frac{18}{2}$$

This simplifies to:

$$s = 9$$

Therefore, the length of the shortest side of the triangle is 9 cm.

Conclusion

In this article, we used algebraic equations to represent the relationships between the sides of a triangle and solve for the length of the shortest side. We started with a given equation representing the perimeter of the triangle and simplified it to isolate the term with $s$. Finally, we solved for the value of $s$ by dividing both sides of the equation by 2. The length of the shortest side of the triangle is 9 cm.

Additional Information

• The longest side of the triangle is $s + 8 = 9 + 8 = 17$ cm.
• The third side of the triangle is 12 cm long.
• The perimeter of the triangle is 38 cm, which means the sum of all three sides is 38 cm.

Mathematical Representation

The equation representing the perimeter of the triangle is:

$$s + (s + 8) + 12 = 38$$

This equation can be simplified to:

$$2s + 20 = 38$$

And further simplified to:

$$2s = 18$$

Finally, solving for $s$ gives us:

$$s = \frac{18}{2}$$

Which simplifies to:

$$s = 9$$

Final Answer

Introduction

In our previous article, we solved a mathematical puzzle involving a triangle with three sides of different lengths. The shortest side was given as $s$ centimeters, while the longest side was 8 cm longer than the shortest side. The third side was 12 cm long, and the perimeter of the triangle was 38 cm. We used algebraic equations to represent the relationships between the sides of the triangle and solve for the value of $s$. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the length of the longest side of the triangle?

A: The longest side of the triangle is $s + 8$ cm. Since we found that $s = 9$ cm, the longest side is $9 + 8 = 17$ cm.

Q: What is the length of the third side of the triangle?

A: The third side of the triangle is given as 12 cm.

Q: What is the perimeter of the triangle?

A: The perimeter of the triangle is 38 cm, which means the sum of all three sides is 38 cm.

Q: How did you simplify the equation?

A: We started by adding the constants on the left-hand side of the equation:

$$s + s + 8 + 12 = 38$$

This simplifies to:

$$2s + 20 = 38$$

Next, we subtracted 20 from both sides of the equation to isolate the term with $s$:

$$2s = 18$$

Q: How did you solve for $s$?

A: We solved for $s$ by dividing both sides of the equation by 2:

$$s = \frac{18}{2}$$

This simplifies to:

$$s = 9$$

Q: What if the perimeter of the triangle was not given?

A: If the perimeter of the triangle was not given, we would need to use the given information to find the perimeter. For example, if we knew the lengths of two sides, we could use the Pythagorean theorem to find the length of the third side.

Q: Can you give an example of a triangle with a different perimeter?

A: Yes, let's say the perimeter of the triangle is 50 cm. We can use the same equation to find the length of the shortest side:

$$s + (s + 8) + 12 = 50$$

Simplifying the equation, we get:

$$2s + 20 = 50$$

Subtracting 20 from both sides, we get:

$$2s = 30$$

Dividing both sides by 2, we get:

$$s = 15$$

Therefore, the length of the shortest side of the triangle is 15 cm.

Conclusion

In this article, we answered some frequently asked questions related to the problem of finding the length of the shortest side of a triangle. We used algebraic equations to represent the relationships between the sides of the triangle and solve for the value of $s$. We also provided examples of triangles with different perimeters.

Additional Information

• The longest side of the triangle is $s + 8 = 9 + 8 = 17$ cm.
• The third side of the triangle is 12 cm long.
• The perimeter of the triangle is 38 cm, which means the sum of all three sides is 38 cm.

Mathematical Representation

The equation representing the perimeter of the triangle is:

$$s + (s + 8) + 12 = 38$$

This equation can be simplified to:

$$2s + 20 = 38$$

And further simplified to:

$$2s = 18$$

Finally, solving for $s$ gives us:

$$s = \frac{18}{2}$$

Which simplifies to:

$$s = 9$$

Final Answer

The length of the shortest side of the triangle is 9 cm.