In triangle PQR, angle QPR is acute, PQ = 10 cm and PR = 14 cm.
(a) The area of triangle PQR is 48 cm². Calculate angle QPR and show that it rounds to 43.3°, correct to 1 decimal place.
Answer (2)
The angle QPR in triangle PQR can be calculated using the area formula to find that \angle QPR \approx 43.3^\circ when rounded to one decimal place.
;
Use the formula for the area of a triangle: A re a = 2 1 ab sin ( C ) .
Substitute the given values: 48 = 2 1 × 10 × 14 × sin ( ∠ QPR ) .
Solve for sin ( ∠ QPR ) : sin ( ∠ QPR ) = 70 48 .
Calculate ∠ QPR using the inverse sine function and round to one decimal place: ∠ QPR ≈ 43. 3 ∘ .
Explanation
Problem Analysis We are given a triangle PQR with sides PQ = 10 cm, PR = 14 cm, and the area of the triangle is 48 cm 2 . We need to find the angle QPR and show that it rounds to 43. 3 ∘ to one decimal place.
Applying the Area Formula We can use the formula for the area of a triangle, which is given by: A re a = 2 1 ab sin ( C ) In our case, A re a = 48 , a = PQ = 10 , b = PR = 14 , and C = ∠ QPR . So we have: 48 = 2 1 × 10 × 14 × sin ( ∠ QPR ) 48 = 70 × sin ( ∠ QPR ) Now, we solve for sin ( ∠ QPR ) :
sin ( ∠ QPR ) = 70 48 = 35 24
Calculating the Angle To find the angle QPR , we take the inverse sine (arcsin) of 35 24 :
∠ QPR = arcsin ( 35 24 ) Using a calculator, we find the value of ∠ QPR in degrees: ∠ QPR ≈ 43.291 8 ∘
Rounding to One Decimal Place Rounding the angle to one decimal place, we get: ∠ QPR ≈ 43. 3 ∘ Thus, we have shown that the angle QPR rounds to 43. 3 ∘ to one decimal place.
Final Answer Therefore, the angle QPR is approximately 43. 3 ∘ .
Examples
Understanding triangle areas is crucial in various fields, such as architecture and surveying. For instance, when designing a building, architects use trigonometry to calculate the area of triangular sections, ensuring efficient use of space and materials. Similarly, surveyors use these principles to determine land areas, which is essential for property management and construction planning. Knowing how to calculate angles and areas in triangles allows professionals to create accurate and safe designs.
