Construct the parallelogram PQRS, where |PQ| = 9 cm, |PS| = 5 cm and |∠SPQ| = 65° - Leaving Cert Mathematics - Question 6 - 2019

To find the area (A) of the parallelogram, you can use the formula:

$A = ab \sin C$

Where:

• a = length of PS = 5 cm
• b = length of PQ = 9 cm
• C = angle SPQ = 65°

Calculating the area:

$A = 9 \times 5 \times \sin(65°)$ $A \approx 9 \times 5 \times 0.906307787 = 40.78 \text{ cm}^2$

Therefore, the area of the parallelogram PQRS is approximately 40.78 cm².

To find α and β:

1. Since O is the center, angle α is inscribed and is half of the arc. Thus, if angle at the circumference is 28°, then: $\alpha = 28°$

2. For β, since it subtends the same arc as α, by the inscribed angle theorem: $\beta = 2 \times 28° = 52°$

Final values:

• α = 28°
• β = 52°