A boat's triangular sail, $P Q R$, is such that the side $PQ =9 cm, PR =10 cm, \measuredangle Q =90^{\circ}$ and $\measuredangle RPQ =25^{\circ}$. Calculate the area of the triangle, correct to 2d.p.
$\begin{array}{l}
{\left[\sin 25^{\circ}=0.4226, \cos 25^{\circ}=0.9063\right.}\\
\left.\tan 25^{\circ}=0.4663\right]
\end{array}$
Answer (2)
The area of the triangular sail PQR is calculated as 18.89 cm². This is determined using the formula for the area of a triangle, involving the base and the height. The height is found using the tangent function based on known angles.
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Identify the given information: a right-angled triangle PQR with PQ = 9 cm, ∠ RPQ = 2 5 ∘ , and ∠ Q = 9 0 ∘ .
Use the tangent function to find the length of side QR : QR = 9 × tan ( 2 5 ∘ ) = 9 × 0.4663 = 4.1967 cm.
Calculate the area of the triangle using the formula A re a = 2 1 × ba se × h e i g h t = 2 1 × 9 × 4.1967 = 18.88515 cm 2 .
Round the area to two decimal places: 18.89 cm 2 .
Explanation
Problem Analysis We are given a right-angled triangle PQR with ∠ Q = 9 0 ∘ , PQ = 9 cm, PR = 10 cm, and ∠ RPQ = 2 5 ∘ . We need to find the area of this triangle.
Area Formula The area of a triangle is given by the formula: A re a = 2 1 × ba se × h e i g h t
In this case, we can consider PQ as the base and QR as the height. We know PQ = 9 cm, but we need to find QR .
Finding the Height To find QR , we can use the tangent function, since we know ∠ RPQ and PQ :
tan ( ∠ RPQ ) = PQ QR tan ( 2 5 ∘ ) = 9 QR QR = 9 × tan ( 2 5 ∘ ) We are given that tan ( 2 5 ∘ ) = 0.4663 , so: QR = 9 × 0.4663 = 4.1967 cm
Calculating the Area Now we can calculate the area of the triangle: A re a = 2 1 × PQ × QR A re a = 2 1 × 9 × 4.1967 A re a = 0.5 × 9 × 4.1967 = 18.88515 cm 2
Final Answer We need to round the area to 2 decimal places: A re a ≈ 18.89 cm 2
Examples
Triangular sails on boats are a classic example of how triangles are used in real life. Calculating the area of a sail helps determine how much wind it can catch, which directly affects the boat's speed and efficiency. Knowing the area allows sailors to optimize their sail size for different wind conditions, ensuring the best performance. This calculation is also crucial in designing sails that are both effective and structurally sound, preventing tears or damage in strong winds.
