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Answer:
Step-by-step explanation:
we know that
The area of a circle is equal to
In this problem we have
Substitute and find the area
Remember that
subtends the area of complete circle
so
by proportion
Find the area of the shaded regions
The central angle of the shaded regions is equal to
Answer:
(a) The area of the triangle is approximately 39.0223 cm²
(b) ∠SQR is approximately 55.582°
Step-by-step explanation:
(a) By sin rule, we have;
SQ/(sin(∠SPQ)) = PQ/(sin(∠PSQ)), which gives;
5.4/(sin(52°)) = 6.8/(sin(∠PSQ))
∴ (sin(∠PSQ)) = (6.8/5.4) × (sin(52°)) ≈ 0.9923
∠PSQ = sin⁻¹(0.9923) ≈ 82.88976°
Similarly, we have;
5.4/(sin(52°)) = SP/(sin(180 - 52 - 82.88976))
Where, 180 - 52 - 82.88976 = ∠PQS = 45.11024
SP = 5.4/(sin(52°))×(sin(180 - 52 - 82.88976)) ≈ 4.8549
Given that RS : SP = 2 : 1, we have;
RS = 2 × SP = 2 × 4.8549 ≈ 9.7098
We have by cosine rule, ² =
² +
² - 2 ×
×
× cos(∠QSR)
∠QSR and ∠PSQ are supplementary angles, therefore;
∠QSR = 180° - ∠PSQ = 180° - 82.88976° = 97.11024°
∠QSR = 97.11024°
Therefore;
² = 5.4² + 9.7098² - 2 × 5.4×9.7098× cos(97.11024)
² ≈ 136.42
= √(136.42) ≈ 11.6799
The area of the triangle = 1/2 × ×
× sin(∠SPQ)
By substituting the values, we have;
1/2 × ×
× sin(∠SPQ)
1/2 × 6.8 × (4.8549 + 9.7098) × sin(52°) ≈ 39.0223 cm²
The area of the triangle ≈ 39.0223 cm²
(b) By sin rule, we have;
/(sin(∠SQR)) =
/(sin(∠QSR))
By substituting, we have;
9.7098/(sin(∠SQR)) = 11.6799/(sin(97.11024))
sin(∠SQR) = 9.7098/(11.6799/(sin(97.11024))) ≈ 0.82493
∠SQR = sin⁻¹(0.82493) ≈ 55.582°.