Problem 215
Question
In the following exercises, solve using triangle properties. The perimeter of a triangular pool is 36 yards. The lengths of two sides are 10 yards and 15 yards. How long is the third side?
Step-by-Step Solution
Verified Answer
The third side is 11 yards long.
1Step 1: Understand the problem
A triangle's perimeter is the sum of its three sides. Given the perimeter is 36 yards and two sides are 10 yards and 15 yards, the task is to find the length of the third side.
2Step 2: Write down the perimeter formula
The perimeter of a triangle can be found using the formula: \[ \text{Perimeter} = a + b + c \]where a, b, and c are the lengths of the three sides.
3Step 3: Substitute the known values
Substitute the given values into the formula: \[ 36 = 10 + 15 + c \]
4Step 4: Solve for the unknown side
To find the unknown side, isolate c by performing the subtraction: \[ c = 36 - 10 - 15 \]
5Step 5: Calculate the length of the third side
Perform the calculation: \[ c = 36 - 25 = 11 \]
Key Concepts
Perimeter FormulaTriangle PropertiesSolving Equations
Perimeter Formula
The perimeter of a shape is the total distance around its boundary. For a triangle, the perimeter is the sum of all its three sides. This is given by the formula:
\[ \text{Perimeter} = a + b + c \]
where \[a, b, \text{and} \ c \] are the lengths of the triangle's sides.
Knowing how to use this formula is crucial for solving problems involving the perimeter of triangles.
When you are given the perimeter and some side lengths, you can rearrange the formula to solve for the unknown side.
\[ \text{Perimeter} = a + b + c \]
where \[a, b, \text{and} \ c \] are the lengths of the triangle's sides.
Knowing how to use this formula is crucial for solving problems involving the perimeter of triangles.
When you are given the perimeter and some side lengths, you can rearrange the formula to solve for the unknown side.
Triangle Properties
Triangles have several important properties that are useful in geometric calculations.
Here’s what you need to know:
* **Sides**: A triangle has three sides.
* **Sum of the lengths of sides**: The sum of any two sides must be greater than the third side.
* **Angles**: The sum of the angles in any triangle always equals 180 degrees.
In our problem, we use the property of the triangle's perimeter to find the unknown side. Given two sides and the perimeter, the length of the third side can be calculated.
Here’s what you need to know:
* **Sides**: A triangle has three sides.
* **Sum of the lengths of sides**: The sum of any two sides must be greater than the third side.
* **Angles**: The sum of the angles in any triangle always equals 180 degrees.
In our problem, we use the property of the triangle's perimeter to find the unknown side. Given two sides and the perimeter, the length of the third side can be calculated.
Solving Equations
Solving for an unknown value is a fundamental skill in algebra and geometry. Let's break down the steps:
* **Identify what you know**: Start with the known values. In our problem, these are the perimeter (36 yards) and the lengths of two sides (10 yards and 15 yards).
* **Set up the equation**: Use the perimeter formula and plug in the known values: \[36 = 10 + 15 + c \]
* **Isolate the unknown**: Rearrange the equation to find the unknown side length, \[c = 36 - 10 - 15 \]
* **Solve**: Perform the subtraction step-by-step: \[36 - 10 = 26 ewline 26 - 15 = 11 \]
Thus, the length of the third side is 11 yards. Solving equations like this helps to systematically find unknown values in geometry.
* **Identify what you know**: Start with the known values. In our problem, these are the perimeter (36 yards) and the lengths of two sides (10 yards and 15 yards).
* **Set up the equation**: Use the perimeter formula and plug in the known values: \[36 = 10 + 15 + c \]
* **Isolate the unknown**: Rearrange the equation to find the unknown side length, \[c = 36 - 10 - 15 \]
* **Solve**: Perform the subtraction step-by-step: \[36 - 10 = 26 ewline 26 - 15 = 11 \]
Thus, the length of the third side is 11 yards. Solving equations like this helps to systematically find unknown values in geometry.
Other exercises in this chapter
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