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PQR and QSR are triangles - Edexcel - GCSE Maths - Question 19 - 2022 - Paper 2
Question 19
PQR and QSR are triangles. Calculate the length of QS. Give your answer correct to 3 significant figures. You must show all your working.
Worked Solution & Example Answer:PQR and QSR are triangles - Edexcel - GCSE Maths - Question 19 - 2022 - Paper 2
Step 1
Calculate the length of QR using the sine rule:
Answer
We can use the sine rule to find the length of QR in triangle PQR. The sine rule states that ( \frac{a}{\sin A} = \frac{b}{\sin B} ). In this case:
- Let ( QR = a )
- Let ( PR = 9.4 , cm )
- Let angle ( P = 27^\circ )
- Let angle ( R = 88^\circ )
Calculating angle Q:
[ Q = 180^\circ - P - R = 180^\circ - 27^\circ - 88^\circ = 65^\circ ]
Now applying the sine rule:
[ \frac{QR}{\sin(27^\circ)} = \frac{9.4}{\sin(65^\circ)} ]
So,
[ QR = \frac{9.4 \times \sin(27^\circ)}{\sin(65^\circ)} ]
Calculating:\n[ QR \approx \frac{9.4 \times 0.454}{0.906} \approx 4.66 , cm ]
Step 2
Use the sine rule to calculate QS:
Answer
In triangle QRS:
- Let ( QS = b )
- ( QR = 4.66 , cm )
- Angle S = 41°
- Angle R = 88°
Calculating angle Q:
[ Q = 180^\circ - S - R = 180^\circ - 41^\circ - 88^\circ = 51^\circ ]
Now applying the sine rule once again:
[ \frac{QS}{\sin(41^\circ)} = \frac{QR}{\sin(51^\circ)} ]
So,
[ QS = \frac{QR \times \sin(41^\circ)}{\sin(51^\circ)} ]
Calculating: [ QS \approx \frac{4.66 \times 0.656}{0.777} \approx 3.94 , cm ]
Step 3