Area of a Triangle using trigonometry
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Area of a Triangle using trigonometry — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionProblem Solving with Non-Right TrianglesConcept Practice
2 marks~3 minCriterion A
A landscape architect is designing a triangular garden plot. The plot has two boundary sides of length AB=7AB = 7 m and BC=9BC = 9 m, with the angle between them measuring ∠ABC=40°\angle ABC = 40°.
a
Deduce which angle must be substituted into the formula Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C when a=7a = 7 m and b=9b = 9 m, and justify your reasoning. [1]
b
Calculate the area of the triangular plot. [1]
Question diagram

Solutions

2QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
4 marks~6 minCriterion C
A landscape architect is designing a triangular garden bed with two fixed side lengths a=9.2a = 9.2 cm and b=12.5b = 12.5 cm (on a scale drawing). The graph below shows how the area of the triangle varies with the included angle CC, using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C.
a
Show that the area formula simplifies to Area=57.5sin⁡C\text{Area} = 57.5\sin C. [1]
b
Explain how the shape of the graph shows that the maximum area occurs at C=90°C = 90°. [2]
c
The architect claims a right-angled triangular bed gives the most efficient use of the fixed side lengths. Calculate the maximum area correct to 3 significant figures, and advise the architect whether this right-angle design should be adopted to maximise the garden bed area. [1]
Question diagram

Solutions

3QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice
4 marks~6 minCriterion D
A surveyor is mapping a triangular plot of land, triangle PQRPQR, where PQ=10PQ = 10 m, PR=14PR = 14 m, and angle Q=40°Q = 40°. The angle at QQ is not the included angle between sides PQPQ and PRPR.

Local planning regulations require any triangular plot submitted for residential development to have an area of at least 6060 m².
a
Use the sine rule to calculate angle RR. Give your answer to one decimal place. [1]
b
Calculate the area of triangle PQRPQR using the formula Area=12absin⁡C\text{Area} = \dfrac{1}{2}ab\sin C. Give your answer to two decimal places. [1]
c
Advise the surveyor whether this plot should be submitted for residential development, referring to the area requirement and the reliability of the result. [2]
Question diagram

Solutions

4QuestionProblem Solving with Non-Right TrianglesAssessment Practice
6 marks~9 minCriterion B
A triangular solar panel frame is being designed with two fixed struts of length 10 cm and 14 cm. The included angle CC between the struts can be adjusted during assembly.
a
Using Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C, calculate the area of the triangle for each angle below. [2]

Angle CC (degrees): 20, 40, 60, 80, 100

Area (cm²): ___, ___, ___, ___, ___
b
Describe the pattern in the areas as CC increases from 20° to 100°, and deduce the angle that produces the maximum area. [2]
c
Justify the engineer's claim that C=90°C = 90° produces the greatest area, using the behaviour of sin⁡C\sin C, and advise the assembly team on how to set the struts and what happens to the enclosed area if they deviate from this angle. [2]

Solutions

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