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Construct the parallelogram P Q R S, where |P Q| = 9 cm, |P S| = 5 cm and |∠S P Q| = 65° - Leaving Cert Mathematics - Question 6 - 2019
Question 6
Construct the parallelogram P Q R S, where |P Q| = 9 cm, |P S| = 5 cm and |∠S P Q| = 65°. The point P has been marked in for you. Show all your construction lines, a... show full transcript
Worked Solution & Example Answer:Construct the parallelogram P Q R S, where |P Q| = 9 cm, |P S| = 5 cm and |∠S P Q| = 65° - Leaving Cert Mathematics - Question 6 - 2019
Step 1
Construct the parallelogram P Q R S
Answer
- Begin by placing the point P on a straight horizontal line.
- Using a protractor, measure an angle of 65° from point P and mark this position as S.
- From point P, measure a length of 5 cm and mark point S.
- From point S, draw a line parallel to P Q for a length of 9 cm to locate point R.
- Finally, connect points Q and R to complete the parallelogram. Ensure to label all points clearly.
Step 2
Find the area of the parallelogram P Q R S
Answer
The area A of a parallelogram can be calculated using the formula:
where:
- b is the base (length |P Q| = 9 cm)
- h is the height, which can be calculated as:
where and |P S| = 5 cm.
Calculating height:
\approx 5 imes 0.9063 \ \\ \approx 4.5319 ext{ cm}$$ Now substituting back into the area formula: $$A = 9 imes 4.5319 \ \\ \approx 40.7871 ext{ cm}^2$$ Rounding to 2 decimal places, the area is approximately 40.79 cm².Step 3
Find the value of α and the value of β
Answer
From the diagram, we know:
- The angle ∠WXY at the circle is related to angle ∠WOS at the center: ∠WXY = 1/2 ∠WOS.
Since ∠WXY = 52°, we can deduce:
Thus, .
- To find α, using the property of a circle, angles subtended by the same chord are equal:
Since angle at the center is double that at the circumference:
Therefore, we have:
- α = 52°
- β = 26°.