In the diagram below, ABCD is a quadrilateral with diagonal AC drawn - NSC Mathematics - Question 7 - 2017 - Paper 2
Question 7
In the diagram below, ABCD is a quadrilateral with diagonal AC drawn.
AB = BC = 17 m
AD = 13 m
∠D = 75°
∠B = 105°
Calculate:
7.1 The area of Δ ABC.
7.2 The length... show full transcript
Worked Solution & Example Answer:In the diagram below, ABCD is a quadrilateral with diagonal AC drawn - NSC Mathematics - Question 7 - 2017 - Paper 2
Step 1
7.1 The area of Δ ABC.
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Answer
To calculate the area of triangle ABC, we can use the formula:
Area=21⋅AB⋅BC⋅sin(B)
Substituting the values:
Area=21⋅17⋅17⋅sin(105°)
Calculating gives:
Area≈139.58m2
Step 2
7.2 The length of AC.
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Answer
To find the length of AC, we can use the cosine rule:
AC2=AB2+BC2−2⋅AB⋅BC⋅cos(B)
Substituting the known values:
AC2=172+172−2⋅17⋅17⋅cos(105°)
Calculating this yields:
AC≈26.97m
Step 3
7.3 The size of ∠CD.
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Answer
Using the sine rule in triangle ACD, we have:
ACsin(∠ACD)=ADsin(∠CAD)
Substituting the known values:
26.97sin(∠ACD)=13sin(75°)
Solving this gives:
∠ACD≈26.97°
Step 4
7.4 Give a reason why ABCD is a cyclic quadrilateral.
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Answer
A necessary condition for a quadrilateral to be cyclic is that the opposite angles must sum to 180°. Here:
