The sides of a quadrilateral are in the ratio 2:3:5:8. The difference between the shortest side and the longest side is 7.5 cm. Work out the perimeter of the quadrilateral.

Steps to Solve

1. Set Variables for Sides

Let the lengths of the sides of the quadrilateral be represented by the variables based on the given ratio:

• Shortest side ( = 2x )
• Second side ( = 3x )
• Third side ( = 5x )
• Longest side ( = 8x )

2. Set Up the Equation Using the Given Information

According to the problem, the difference between the longest side and the shortest side is 7.5 cm. We can express this relationship with the equation:

$$ 8x - 2x = 7.5 $$

3. Solve for (x)

Simplifying the equation gives us:

$$ 6x = 7.5 $$

To find (x), divide both sides by 6:

$$ x = \frac{7.5}{6} = 1.25 $$

4. Calculate the Lengths of Each Side

Now, substitute (x) back into the expressions for each side:

• Shortest side: (2x = 2 \times 1.25 = 2.5 , \text{cm})
• Second side: (3x = 3 \times 1.25 = 3.75 , \text{cm})
• Third side: (5x = 5 \times 1.25 = 6.25 , \text{cm})
• Longest side: (8x = 8 \times 1.25 = 10 , \text{cm})

5. Calculate the Perimeter

Add all the sides together to find the perimeter (P):

$$ P = 2.5 + 3.75 + 6.25 + 10 $$

Calculating this gives:

$$ P = 22.5 , \text{cm} $$