Answer :
Sure, let's calculate the perimeter of the rectangle step-by-step.
### Step 1: Understanding the problem
We are given a rectangle with:
- Diagonal \( d = 15 \) cm
- Length \( l = 12 \) cm
We need to find the perimeter of the rectangle.
### Step 2: Use the Pythagorean Theorem
In a rectangle, the diagonal forms a right-angled triangle with the length and the width. According to the Pythagorean theorem:
[tex]\[ d^2 = l^2 + w^2 \][/tex]
where \( w \) is the width of the rectangle.
Rearranging for \( w^2 \):
[tex]\[ w^2 = d^2 - l^2 \][/tex]
### Step 3: Substitute the known values
Substituting \( d = 15 \) cm and \( l = 12 \) cm, we get:
[tex]\[ w^2 = 15^2 - 12^2 \][/tex]
[tex]\[ w^2 = 225 - 144 \][/tex]
[tex]\[ w^2 = 81 \][/tex]
### Step 4: Calculate width \( w \)
Taking the square root of both sides:
[tex]\[ w = \sqrt{81} \][/tex]
[tex]\[ w = 9 \][/tex]
Now we have:
- Length \( l = 12 \) cm
- Width \( w = 9 \) cm
### Step 5: Calculate the perimeter
The formula for the perimeter \( P \) of a rectangle is given by:
[tex]\[ P = 2(l + w) \][/tex]
Substituting the values of \( l \) and \( w \):
[tex]\[ P = 2(12 + 9) \][/tex]
[tex]\[ P = 2(21) \][/tex]
[tex]\[ P = 42 \][/tex]
### Final Answer
The perimeter of the rectangle is [tex]\( 42 \)[/tex] cm.
### Step 1: Understanding the problem
We are given a rectangle with:
- Diagonal \( d = 15 \) cm
- Length \( l = 12 \) cm
We need to find the perimeter of the rectangle.
### Step 2: Use the Pythagorean Theorem
In a rectangle, the diagonal forms a right-angled triangle with the length and the width. According to the Pythagorean theorem:
[tex]\[ d^2 = l^2 + w^2 \][/tex]
where \( w \) is the width of the rectangle.
Rearranging for \( w^2 \):
[tex]\[ w^2 = d^2 - l^2 \][/tex]
### Step 3: Substitute the known values
Substituting \( d = 15 \) cm and \( l = 12 \) cm, we get:
[tex]\[ w^2 = 15^2 - 12^2 \][/tex]
[tex]\[ w^2 = 225 - 144 \][/tex]
[tex]\[ w^2 = 81 \][/tex]
### Step 4: Calculate width \( w \)
Taking the square root of both sides:
[tex]\[ w = \sqrt{81} \][/tex]
[tex]\[ w = 9 \][/tex]
Now we have:
- Length \( l = 12 \) cm
- Width \( w = 9 \) cm
### Step 5: Calculate the perimeter
The formula for the perimeter \( P \) of a rectangle is given by:
[tex]\[ P = 2(l + w) \][/tex]
Substituting the values of \( l \) and \( w \):
[tex]\[ P = 2(12 + 9) \][/tex]
[tex]\[ P = 2(21) \][/tex]
[tex]\[ P = 42 \][/tex]
### Final Answer
The perimeter of the rectangle is [tex]\( 42 \)[/tex] cm.
