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In the diagram, A, B(-6 ; -5) and C(8 ; -4) are points in the Cartesian plane - NSC Mathematics - Question 3 - 2017 - Paper 2
Question 3
In the diagram, A, B(-6 ; -5) and C(8 ; -4) are points in the Cartesian plane. F(3 ; \frac{3}{2}) and G are points on line AC such that AF = FG. E is the x-intercept... show full transcript
Worked Solution & Example Answer:In the diagram, A, B(-6 ; -5) and C(8 ; -4) are points in the Cartesian plane - NSC Mathematics - Question 3 - 2017 - Paper 2
Step 1
3.1.1 The equation of AC in the form y = mx + c
Answer
To find the equation of line AC, we first calculate the slope (m) using the coordinates of points A and C:
Using point A(2, 5) to find the y-intercept (c) of the line AC:
[ y - y_1 = m(x - x_1) ] [ y - 5 = -\frac{3}{2}(x - 2) ]
Expanding this gives:
[ y - 5 = -\frac{3}{2}x + 3 ] [ y = -\frac{3}{2}x + 8 ]
Thus, the equation of AC is:\n[ y = -\frac{3}{2}x + 8 ]
Step 2
3.1.2 The coordinates of G if the equation of BG is 7x - 10y = 8
Answer
To find the coordinates of G, we first need the coordinates of point B which is (-6, -5). We will rewrite the equation of line BG in slope-intercept form:
Starting from ( 7x - 10y = 8 ): [ -10y = -7x + 8 ] [ y = \frac{7}{10}x - \frac{8}{10} ]
Finding the intersection of lines AC and BG will give us the coordinates of G:
Substituting the equation of AC into BG: [ -\frac{3}{2}x + 8 = \frac{7}{10}x - \frac{8}{10} ]
Clearing the fractions by multiplying through by 20: [ -30x + 160 = 14x - 16 ] [ 44x = 176 ] [ x = 4 ]
Now substituting ( x = 4 ) back into either equation to find y: [ y = -\frac{3}{2}(4) + 8 = 2 ]
Thus, the coordinates of G are (4, 2).
Step 3
3.2 Show by calculation that the coordinates of A is (2 ; 5)
Answer
To verify the coordinates of A, we can check the midpoint E of segment AB:
Using the midpoint formula: [ E = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-6 + 2}{2}, \frac{-5 + 5}{2} \right) = (\frac{-4}{2}, 0) = (-2, 0) ]
Since we know point A should be (2, 5), we can verify the coordinates based on this midpoint calculation. Thus, the coordinates (2, 5) accurately describe point A.
Step 4
3.3 Prove that EF || BG
Answer
To prove that segments EF and BG are parallel, we compare their slopes. Using the coordinates of points E and F:
Calculate the slope of EF: [ m_{EF} = \frac{y_2 - y_1}{x_2 - x_1} ] With E (–2, 0) and F (3, \frac{3}{2}): [ m_{EF} = \frac{\frac{3}{2} - 0}{3 - (-2)} = \frac{\frac{3}{2}}{5} = \frac{3}{10} ]
Now, using the coordinates of points B and G to calculate the slope of BG: [ m_{BG} = \frac{-5 - 2}{-6 - 4} = \frac{-7}{-10} = \frac{7}{10} ]
As the slopes are equal, we conclude that EF is indeed parallel to BG.
Step 5
3.4 ABCD is a parallelogram with D in the first quadrant. Calculate the coordinates of D.
Answer
In parallelogram ABCD, the diagonals bisect each other, which means D's coordinates can be found as follows:
Given midpoints: [ \text{Midpoint AC} = \left( \frac{2+8}{2}, \frac{5 + (-4)}{2} \right) = (5, \frac{1}{2}) ] [ \text{Midpoint BD} = \left( \frac{-6 + x}{2}, \frac{-5 + y}{2} \right) ]
Since these midpoints are equal, we can equate the x and y coordinates: [ \frac{-6 + x}{2} = 5 \quad \text{and} \quad \frac{-5 + y}{2} = \frac{1}{2} ]
Solving for x: [ -6 + x = 10 \Rightarrow x = 16 ]
And for y: [ -5 + y = 1 \Rightarrow y = 6 ]
Thus, the coordinates of D are (16, 6).