Absolutely, let's solve this step-by-step.
### Step 1: Define Variables
First, let's denote the length of the rectangle by [tex]\( l \)[/tex] and the width by [tex]\( w \)[/tex]. Given that the adjacent sides are in the ratio 5:12, we can express the length and width in terms of a single variable [tex]\( x \)[/tex]:
[tex]\[ l = 5x \][/tex]
[tex]\[ w = 12x \][/tex]
### Step 2: Use the Perimeter Formula
The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula:
[tex]\[ P = 2(l + w) \][/tex]
Given the perimeter is 34 cm, we substitute the expressions for [tex]\( l \)[/tex] and [tex]\( w \)[/tex]:
[tex]\[ 34 = 2(5x + 12x) \][/tex]
[tex]\[ 34 = 2(17x) \][/tex]
[tex]\[ 34 = 34x \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Solving this equation for [tex]\( x \)[/tex]:
[tex]\[ 34 = 34x \][/tex]
[tex]\[ x = 1 \][/tex]
### Step 4: Find Length and Width
Now that we have [tex]\( x \)[/tex], we can find the length and the width:
[tex]\[ l = 5x = 5 \times 1 = 5 \text{ cm} \][/tex]
[tex]\[ w = 12x = 12 \times 1 = 12 \text{ cm} \][/tex]
### Step 5: Calculate the Diagonal
The length of the diagonal [tex]\( d \)[/tex] of a rectangle can be found using the Pythagorean theorem:
[tex]\[ d = \sqrt{l^2 + w^2} \][/tex]
Substitute the values of [tex]\( l \)[/tex] and [tex]\( w \)[/tex] we found:
[tex]\[ d = \sqrt{5^2 + 12^2} \][/tex]
[tex]\[ d = \sqrt{25 + 144} \][/tex]
[tex]\[ d = \sqrt{169} \][/tex]
[tex]\[ d = 13 \text{ cm} \][/tex]
### Answer
Thus, the length of the diagonal of the rectangle is [tex]\( 13 \text{ cm} \)[/tex].
