Math / Geometry

QuestionThe perimeter of a triangle is 107 m . The second side is three times as long as the first side. The third side is 5 m shorter than than the second side. How long is each side.
Determine the length of each side of the triangle. The first side is $\square$ m
The second side is $\square$ m.
The third side is $\square$ m

Studdy Solution

STEP 1

What is this asking? We need to find the lengths of the three sides of a triangle when we know the total length around the triangle and how the side lengths relate to each other. Watch out! Don't forget that the perimeter is the sum of all sides!
Also, make sure to clearly define what your variables represent to avoid confusion.

STEP 2

1. Set up variables and equations
2. Solve for the first side
3. Calculate other sides

STEP 3

Let's define our variables!
We'll call the length of the first side $x$ .
Since the second side is three times as long as the first side, we can represent its length as $3x$ .
The third side is 5 m shorter than the second side, so its length is $3x - 5$ .

STEP 4

Now, let's build our equation!
We know the perimeter is the sum of all the sides, and it's given as 107 m.
So, we can write the equation: $x + 3x + (3x - 5) = 107$ .
This equation says the sum of the lengths of the three sides equals the perimeter.

STEP 5

Let's simplify our equation!
Combining the $x$ terms, we get $7x - 5 = 107$ .

STEP 6

To isolate the $x$ term, we'll add 5 to both sides of the equation: $7x - 5 + 5 = 107 + 5$ , which simplifies to $7x = 112$ .
We're adding 5 to both sides to make the left side simpler.

STEP 7

Now, we'll divide both sides by 7 to solve for $x$ : $\frac{7x}{7} = \frac{112}{7}$ .
This gives us $x = 16$ .
So, the length of the first side is 16 m!
We divide by 7 to turn the $7x$ into a single $x$ .

STEP 8

We know the second side is three times the length of the first side, so it's $3 \cdot 16 = 48$ .
The second side is 48 m long!

STEP 9

The third side is 5 m shorter than the second side, so it's $48 - 5 = 43$ .
The third side is 43 m long!

STEP 10

The lengths of the sides are: First side: 16 m Second side: 48 m Third side: 43 m

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Studdy Solution

STEP 1

STEP 2

1. Set up variables and equations
2. Solve for the first side
3. Calculate other sides

STEP 3

Let's define our variables!
We'll call the length of the first side $x$ .
Since the second side is three times as long as the first side, we can represent its length as $3x$ .
The third side is 5 m shorter than the second side, so its length is $3x - 5$ .

STEP 4

Now, let's build our equation!
We know the perimeter is the sum of all the sides, and it's given as 107 m.
So, we can write the equation: $x + 3x + (3x - 5) = 107$ .
This equation says the sum of the lengths of the three sides equals the perimeter.

STEP 5

Let's simplify our equation!
Combining the $x$ terms, we get $7x - 5 = 107$ .

STEP 6

To isolate the $x$ term, we'll add 5 to both sides of the equation: $7x - 5 + 5 = 107 + 5$ , which simplifies to $7x = 112$ .
We're adding 5 to both sides to make the left side simpler.

STEP 7

Now, we'll divide both sides by 7 to solve for $x$ : $\frac{7x}{7} = \frac{112}{7}$ .
This gives us $x = 16$ .
So, the length of the first side is 16 m!
We divide by 7 to turn the $7x$ into a single $x$ .

STEP 8

We know the second side is three times the length of the first side, so it's $3 \cdot 16 = 48$ .
The second side is 48 m long!

STEP 9

The third side is 5 m shorter than the second side, so it's $48 - 5 = 43$ .
The third side is 43 m long!

STEP 10

The lengths of the sides are: First side: 16 m Second side: 48 m Third side: 43 m

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Studdy Solution

STEP 1

STEP 2

1. Set up variables and equations2. Solve for the first side3. Calculate other sides

STEP 3

Let's **define** our variables!We'll call the length of the first side xxx.Since the second side is **three times** as long as the first side, we can represent its length as 3x3x3x.The third side is **5 m shorter** than the second side, so its length is 3x−53x - 53x−5.

STEP 4

Now, let's **build** our equation!We know the **perimeter** is the sum of all the sides, and it's given as **107 m**.So, we can write the equation: x+3x+(3x−5)=107x + 3x + (3x - 5) = 107x+3x+(3x−5)=107.This equation says the sum of the lengths of the three sides equals the perimeter.

STEP 5

Let's **simplify** our equation!Combining the xxx terms, we get 7x−5=1077x - 5 = 1077x−5=107.

STEP 6

To **isolate** the xxx term, we'll **add 5** to both sides of the equation: 7x−5+5=107+57x - 5 + 5 = 107 + 57x−5+5=107+5, which simplifies to 7x=1127x = 1127x=112.We're adding 5 to both sides to make the left side simpler.

STEP 7

Now, we'll **divide** both sides by **7** to solve for xxx: 7x7=1127\frac{7x}{7} = \frac{112}{7}77x​=7112​.This gives us x=16x = 16x=16.So, the length of the first side is **16 m**!We divide by 7 to turn the 7x7x7x into a single xxx.

STEP 8

We know the second side is **three times** the length of the first side, so it's 3⋅16=483 \cdot 16 = 483⋅16=48.The second side is **48 m** long!

STEP 9

The third side is **5 m shorter** than the second side, so it's 48−5=4348 - 5 = 4348−5=43.The third side is **43 m** long!

STEP 10

The lengths of the sides are: First side: **16 m** Second side: **48 m** Third side: **43 m**

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Studdy solves anything!

Start learning now

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1. Set up variables and equations
2. Solve for the first side
3. Calculate other sides

Let's define our variables!
We'll call the length of the first side $x$ .
Since the second side is three times as long as the first side, we can represent its length as $3x$ .
The third side is 5 m shorter than the second side, so its length is $3x - 5$ .

Now, let's build our equation!
We know the perimeter is the sum of all the sides, and it's given as 107 m.
So, we can write the equation: $x + 3x + (3x - 5) = 107$ .
This equation says the sum of the lengths of the three sides equals the perimeter.

Let's simplify our equation!
Combining the $x$ terms, we get $7x - 5 = 107$ .

To isolate the $x$ term, we'll add 5 to both sides of the equation: $7x - 5 + 5 = 107 + 5$ , which simplifies to $7x = 112$ .
We're adding 5 to both sides to make the left side simpler.

Now, we'll divide both sides by 7 to solve for $x$ : $\frac{7x}{7} = \frac{112}{7}$ .
This gives us $x = 16$ .
So, the length of the first side is 16 m!
We divide by 7 to turn the $7x$ into a single $x$ .

We know the second side is three times the length of the first side, so it's $3 \cdot 16 = 48$ .
The second side is 48 m long!

The third side is 5 m shorter than the second side, so it's $48 - 5 = 43$ .
The third side is 43 m long!

The lengths of the sides are: First side: 16 m Second side: 48 m Third side: 43 m