A boat's triangular sail, PQR, 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[\operatorname{Sin} 25^{\circ}=0.4226, \operatorname{Cos} 25^{\circ}=0.9063\right.\
\left.\tan 25^{\circ}=0.4663\right]}
$\end{array}
A. $27.43 cm^2$
B. $19.02 cm^2$
C. $18.41 cm^2$
D. $16.30 cm^2
Answer (2)
The area of triangle PQR can be calculated as 18.89 cm 2 , based on provided measurements. This area does not match any of the given multiple-choice answers. Further investigation into the problem might be necessary to reconcile this difference.
;
Use the tangent function to find the length of side QR: QR = PQ × t an ( 2 5 ∘ ) = 9 × 0.4663 = 4.1967 .
Calculate the area of the triangle using the formula: A re a = 2 1 × ba se × h e i g h t = 2 1 × PQ × QR = 2 1 × 9 × 4.1967 = 18.88515 .
Round the area to 2 decimal places: A re a ≈ 18.89 c m 2 .
The area of the triangle PQR is: 18.89 c m 2 .
Explanation
Problem Analysis We are given a triangle PQR with PQ = 9 cm, PR = 10 cm, ∠ PQR = 9 0 ∘ , and ∠ RPQ = 2 5 ∘ . We need to find the area of the triangle.
Find QR First, we need to find the length of side QR. Since we have a right-angled triangle, we can use the tangent function: t an ( 2 5 ∘ ) = PQ QR We are given that t an ( 2 5 ∘ ) = 0.4663 and PQ = 9 cm.
Calculate QR Now we can calculate QR: QR = PQ × t an ( 2 5 ∘ ) = 9 × 0.4663 = 4.1967
Calculate Area Next, we calculate the area of the triangle using the formula: A re a = 2 1 × ba se × h e i g h t = 2 1 × PQ × QR Substituting the values of PQ and QR: A re a = 2 1 × 9 × 4.1967 = 18.88515
Round to 2 d.p. Finally, we round the area to 2 decimal places: A re a ≈ 18.89 c m 2
Final Answer The area of the triangle PQR is approximately 18.89 c m 2 .
Examples
Triangular sails are a classic example of how triangles are used in real life. Calculating the area of a sail is crucial for determining the amount of wind it can catch, which directly affects the boat's speed and maneuverability. Understanding the area allows sailors to optimize their sail size for different wind conditions, ensuring efficient sailing. This principle extends to various other applications, such as designing kites, awnings, and even architectural structures where triangular shapes are employed.
