The diagram below shows the right-angled triangle ABC, which is used in the logo for a company called Deane Construction Limited (DCL) - Leaving Cert Mathematics - Question 7 - 2015

|OB| is measured to be 15 cm and |OQ| is measured to be 22.5 cm.

The scale factor k can be calculated using the formula:

$k = \frac{|OQ|}{|OB|} = \frac{22.5 \text{ cm}}{15 \text{ cm}} = 1.5.$

The area of triangle ABC is given as 7.5 cm². To find the area of triangle PQR, we use the formula:

$\text{Area of PQR} = \text{Area of ABC} \times k^2 = 7.5 \text{ cm}^2 \times (1.5)^2 = 7.5 \times 2.25 = 16.875 \text{ cm}^2.$

Using the scale factor, we find |PQ| as follows:

$|PQ| = |AB| \times k = 5 \text{ cm} \times 1.5 = 7.5 \text{ cm}.$

To find |BC|, we rearrange the scale factor relationship:

$|BC| = |QR| \div k = 8.7 \text{ cm} \div 1.5 = 5.8 \text{ cm}.$

Since the triangles ABC and PQR are similar (due to the enlargement), the corresponding angles are equal. Therefore,

$|∠ABC| = |∠PQR|.$