In the diagram, A(4 ; 2), B(6 ; -4) and C(-2 ; -3) are vertices of AABC - NSC Mathematics - Question 3 - 2022 - Paper 2
Question 3
In the diagram, A(4 ; 2), B(6 ; -4) and C(-2 ; -3) are vertices of AABC. T is the midpoint of CB. The equation of line AC is 5x - 6y = 8. The angle of inclination of... show full transcript
Worked Solution & Example Answer:In the diagram, A(4 ; 2), B(6 ; -4) and C(-2 ; -3) are vertices of AABC - NSC Mathematics - Question 3 - 2022 - Paper 2
Step 1
Gradient of AB
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Answer
The gradient (m) of a line given two points (x1, y1) and (x2, y2) can be calculated using the formula:
m_{AB} = rac{y_2 - y_1}{x_2 - x_1}
For points A(4, 2) and B(6, -4):
m_{AB} = rac{-4 - 2}{6 - 4} = rac{-6}{2} = -3.
Step 2
Size of α
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The angle of inclination α can be found using the tangent of the gradient:
[ \tan(α) = m_{AB} = -3 ]
From this, we find:
[ α = \tan^{-1}(-3) ]
The principal value can be calculated, leading to:
[ α \approx 108.43° ].
Step 3
Coordinates of T
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T, being the midpoint of CB, can be calculated as:
[ T = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]
Using C(-2, -3) and B(6, -4):
[ T = \left( \frac{-2 + 6}{2}, \frac{-3 - 4}{2} \right) = \left( 2, -3.5 \right) ].
Step 4
Coordinates of S
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Answer
To find the intercept S of AC, we use the equation of line AC: 5x - 6y = 8. Setting x = 0 gives:
[ 5(0) - 6y = 8 \Rightarrow -6y = 8 \Rightarrow y = -\frac{4}{3} ].
Thus, S = (0, -\frac{4}{3}).
Step 5
Determine the equation of CD in the form y = mx + c.
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The slope of CD, denoted as m, can be derived from the coordinates of points C(-2, -3) and D(2, -4).
[ m_{CD} = \frac{-4 - (-3)}{2 - (-2)} = \frac{-1}{4} = -\frac{1}{4} ]
Using point-slope form:
y - y_1 = m(x - x_1)
Choosing point D(2, -4):
y + 4 = -\frac{1}{4}(x - 2).
Rearranging:
y = -\frac{1}{4}x - 4 + \frac{1}{2} = -\frac{1}{4}x - \frac{7}{2}.
Step 6
Size of ∠DCA
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To compute the size of ∠DCA, we first need the slopes of lines DC and AC.
Find the slope of AC first (say m_{AC}) and then the angle using:
[ \tan(\theta) = \left| \frac{m_{AC}-m_{CD}}{1 + m_{AC} * m_{CD}} \right|. ]
Where θ is the angle between the two lines.
Step 7
Area of POSC
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The area of polygon POSC can be calculated using the coordinates of points P, O, S, and C.
Using the formula for the area based on vertex coordinates:
[ Area = \frac{1}{2} | x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) |].
Inserting the coordinates accordingly will yield the exact area.
