Solving Triangles with Given Information
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Solving Triangles with Given Information — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionWorking with Two Sides and Included Angle SASConcept Practice
2 marks~3 minCriterion A
A surveyor measures two boundary lines from a fixed point AA: one line AB=5 cmAB = 5 \text{ cm} and another AC=7 cmAC = 7 \text{ cm}, with the angle between them ∠BAC=40°\angle BAC = 40°.

Calculate the length of BCBC. [2]
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Solutions

2QuestionWorking with Two Sides and Included Angle SASAssessment Practice
4 marks~6 minCriterion D
A surveyor is planning a triangular plot of land with two fixed boundary sides a=8a = 8 m and b=11b = 11 m and an adjustable included angle CC. The cosine rule states:

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
a
Calculate the length of side cc when C=60°C = 60° and when C=120°C = 120°. [2]
b
Show that when C=90°C = 90°, the cosine rule reduces to Pythagoras' theorem, and calculate cc. [1]
c
The surveyor needs the third boundary cc to be at least 15 m to meet a planning regulation. Advise the surveyor whether C=90°C = 90° is sufficient, and justify the minimum whole-number angle (in degrees) that should be used. [1]
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Solutions

3QuestionWorking with Two Sides and Included Angle SASAssessment Practice
5 marks~8 minCriterion B
In triangle ABCABC, let BC=aBC = a, AC=bAC = b, AB=cAB = c, and let ∠ACB=C\angle ACB = C.
a
Construct a perpendicular from vertex AA to side BCBC, meeting BCBC at point DD. Label CD=xCD = x, DB=a−xDB = a - x, and height AD=hAD = h. Using right triangle ACDACD, write expressions for xx and hh in terms of bb and CC. [2]
b
Apply the Pythagorean theorem to right triangle ABDABD and substitute your expressions from part (a). Expand and simplify fully. [2]
c
Hence prove that c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C, justifying the final algebraic step that reduces the expression. [1]
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Solutions

4QuestionWorking with Two Sides and Included Angle SASAssessment Practice
6 marks~9 minCriterion C
A surveyor measures two sides of a triangular plot as a=85a = 85 m and b=62b = 62 m, with an included angle of C=47°C = 47°. The cosine rule states:

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
a
Calculate the length of the third side, cc, correct to one decimal place. [2]
b
The surveyor's measurements have possible errors of ±0.5\pm 0.5 m in each side length and ±1°\pm 1° in the angle. Analyse which of these three measurement errors has the greatest impact on the calculated value of cc, and justify your reasoning mathematically. [3]
c
The surveyor's client requires the third side to be shorter than 65 m to meet a planning regulation. Advise the surveyor whether the measurement errors identified in part (b) affect the reliability of the conclusion drawn from part (a). [1]
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Solutions

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