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 lengt... 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
The area of Δ ABC.
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Answer
To find the area of triangle ABC, we can use the area formula:
extArea=21×AB×BC×sin(B)
Substituting the values:
=21×17×17×sin(105°)
Calculating:
=21×17×17×0.9659≈139.58extm2
Step 2
The length of AC.
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Answer
To calculate AC, we will use the cosine rule:
AC2=AB2+BC2−2×AB×BC×cos(B)
Substituting the values:
=172+172−2×17×17×cos(105°)
Calculating:
=289+289−2×17×17×(−0.2588)
This results in:
AC≈26.97extm
Step 3
The size of ∠CD.
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Answer
To find the size of angle ACD, we can use the sine rule:
ACsin(ACD)=ABsin(B)
Rearranging gives:
sin(ACD)=AC×ABsin(B)
Substituting the values:
=26.97×17sin(105°)
Calculating:
=26.97×0.9659≈26.97
Step 4
Give a reason why ABCD is a cyclic quadrilateral.
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Answer
ABCD is a cyclic quadrilateral because the opposite angles add up to 180°. Specifically, ∠B + ∠D = 105° + 75° = 180°.
