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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 9 · 9.5 Coordinate Geometry

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.m=3,c=2m = 3, c = 2
2.2
3.(5,7)(5, 7)
4.5
5.4
6.−1/2-1/2
7.−1,1,3,5-1, 1, 3, 5
8.y=3x−4y = 3x - 4
9.Horizontal: y=7y = 7; vertical: x=−3x = -3
10.x=−2x = -2, y=6y = 6
Silver
11.−3/2-3/2
12.5
13.y=2x+1y = 2x + 1
14.y=2x+2y = 2x + 2
15.y=−x/2+5y = -x/2 + 5
16.(1,−2)(1, -2)
17.y=3x−5y = 3x - 5
18.(a) parallel (b) perpendicular
19.y=−3x+10y = -3x + 10
20.y=4y = 4 (horizontal)
Gold
21.y=(1/3)x+5/3y = (1/3)x + 5/3
22.C=(4,3)C = (4, 3)
23.y=−x+7y = -x + 7
24.x-int (4, 0); y-int (0, 3)
25.Yes (7=77 = 7)
26.y=−(2/3)xy = -(2/3)x
27.5+5+8=185 + 5 + 8 = 18
28.m=−4m = -4
29.F=(9/5,13/5)F = (9/5, 13/5)
30.y=2x+3y = 2x + 3
Platinum
31.(a) (5.5, 7) (b) y=(10/9)x+8/9y = (10/9)x + 8/9
32.Yes — gradients PQ = QR = 2
33.855\dfrac{8\sqrt{5}}{5}
34.y=−(b/a)x+by = -(b/a)x + b (or x/a+y/b=1x/a + y/b = 1)
35.9 sq units
36.Not right-angled (no perpendicular pairs)
37.B=(5,−1)B = (5, -1)
38.AB 0, BC 3, CD 0, DA 3. Parallelogram (parallel sides equal).
39.k=−2/3k = -2/3 (after computation)
40.(a) 3/43/4 (b) xx-int (4, 0); yy-int (0, -3) (c) 12/512/5

Pack B — Answers

Bronze
1.m=−2,c=5m = -2, c = 5
2.2
3.(2,5)(2, 5)
4.10
5.−3-3
6.−1/3-1/3
7.5,4,3,25, 4, 3, 2
8.y=−2x+5y = -2x + 5
9.Horizontal: y=−2y = -2; vertical: x=4x = 4
10.x=4x = 4, y=8y = 8
Silver
11.−2-2
12.525\sqrt{2}
13.y=3x−1y = 3x - 1
14.y=3x−1y = 3x - 1
15.y=−x/3+3y = -x/3 + 3
16.(2,3)(2, 3)
17.y=−2x+2y = -2x + 2
18.Same.
19.y=−2x+8y = -2x + 8
20.x=3x = 3 (vertical)
Gold
21.y=−(1/3)x+1y = -(1/3)x + 1
22.C=(1,4)C = (1, 4)
23.y=(3/2)x−1/2y = (3/2)x - 1/2
24.x-int (4, 0); y-int (0, 10)
25.Yes (5=55 = 5)
26.y=(3/5)x+2/5y = (3/5)x + 2/5
27.10+10+12=3210 + 10 + 12 = 32
28.m=3m = 3
29.Same.
30.y=2x−5y = 2x - 5
Platinum
31.(a) (4, 4) (b) y=(1/5)x+16/5y = (1/5)x + 16/5
32.Same.
33.0
34.Same.
35.Same.
36.Same.
37.B=(5,7)B = (5, 7)
38.Same.
39.Pack B answer: k=−8/3k = -8/3
40.(a) −5/12-5/12 (b) xx-int (12, 0); yy-int (0, 5) (c) 60/1360/13

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 13\tfrac{1}{3} (b) y=13x+53y = \tfrac{1}{3}x + \tfrac{5}{3} (c) y=13xy = \tfrac{1}{3}x
Full working
(a) (4−1)/(7−(−2))=3/9=1/3(4-1)/(7-(-2)) = 3/9 = 1/3.

(b) Use A: 1=(1/3)(−2)+c⇒c=5/31 = (1/3)(-2) + c \Rightarrow c = 5/3. y=x/3+5/3y = x/3 + 5/3.

(c) Same gradient, through origin: y=x/3y = x/3.
2Problem 2
Answer
C=(4,3)C = (4, 3)
Full working
C is 2/3 from A to B. Cx=−2+(2/3)(9)=4C_x = -2 + (2/3)(9) = 4. Cy=1+(2/3)(3)=3C_y = 1 + (2/3)(3) = 3.

Verify: AC=36+4=40AC = \sqrt{36 + 4} = \sqrt{40}; CB=9+1=10CB = \sqrt{9 + 1} = \sqrt{10}. 40/10=2\sqrt{40}/\sqrt{10} = 2 ✓ (ratio 2:1).
3Problem 3
Answer
(a) 10 km (b) (6, 8) (c) y=−(4/3)x+16y = -(4/3)x + 16
Full working
(a) 64+36=10\sqrt{64 + 36} = 10 km.

(b) Midpoint (6, 8).

(c) AB gradient =6/8=3/4= 6/8 = 3/4. Perpendicular gradient −4/3-4/3. Through (6, 8): c=16c = 16.
4Problem 4
Answer
(a) P(3, 0), Q(7, 2), R(5, 5), S(1, 3) (b) PQ ∥ SR (gradient 1/2); QR ∥ PS (gradient −3/2-3/2)
Full working
(a) Compute each midpoint.

(b) PQ gradient = (2−0)/(7−3)=1/2(2-0)/(7-3) = 1/2. SR gradient = (5−3)/(5−1)=1/2(5-3)/(5-1) = 1/2. Equal → parallel. QR: −3/2-3/2; PS: −3/2-3/2. Both pairs parallel → parallelogram (Varignon's theorem).
5Problem 5
Answer
y=−x+7y = -x + 7
Full working
Midpoint (3, 4). AB gradient = 1. Perp gradient = −1-1. Through (3, 4): c=7c = 7.
6Problem 6
Answer
18
Full working
AB =16+9=5= \sqrt{16+9} = 5. BC =16+9=5= \sqrt{16+9} = 5. AC =8= 8. Perimeter = 18.

(Note: an isosceles triangle with the base on the xx-axis.)
7Problem 7
Answer
(a) y=−x/2+7/2y = -x/2 + 7/2 (b) F=(9/5,13/5)F = (9/5, 13/5) (c) 855\tfrac{8\sqrt{5}}{5}
Full working
(a) Perp gradient −1/2-1/2. c=7/2c = 7/2.

(b) Intersect: 2x−1=−x/2+7/2⇒5x=9⇒x=9/5,y=13/52x - 1 = -x/2 + 7/2 \Rightarrow 5x = 9 \Rightarrow x = 9/5, y = 13/5.

(c) Distance =(5−9/5)2+(1−13/5)2=(16/5)2+(−8/5)2=320/25=85/5= \sqrt{(5 - 9/5)^2 + (1 - 13/5)^2} = \sqrt{(16/5)^2 + (-8/5)^2} = \sqrt{320/25} = 8\sqrt{5}/5.
8Problem 8
Answer
(a) m=3/2m = 3/2 (b) m=(3±13)/2m = (3 \pm \sqrt{13})/2
Full working
(a) Parallel: m=3−m⇒m=3/2m = 3 - m \Rightarrow m = 3/2.

(b) Perp: m(3−m)=−1⇒m2−3m−1=0⇒m=(3±13)/2m(3 - m) = -1 \Rightarrow m^2 - 3m - 1 = 0 \Rightarrow m = (3 \pm \sqrt{13})/2.
9Problem 9
Answer
(a) (−1,0)(-1, 0), (5,0)(5, 0), (2,3)(2, 3) (b) 9
Full working
(a) y=x+1y = x+1 meets y=0y = 0 at (−1,0)(-1, 0). y=−x+5y = -x+5 meets y=0y = 0 at (5,0)(5, 0). y=x+1y = x+1 and y=−x+5y = -x+5: x+1=−x+5⇒x=2,y=3x + 1 = -x + 5 \Rightarrow x = 2, y = 3.

(b) Base 6 (from −1-1 to 55 on xx-axis), height 3. Area =9= 9.
10Problem 10
Answer
AB ∥ DC (both horizontal). AD ∥ BC (gradient 3/2)
Full working
AB gradient = 0; DC gradient = 0. AD: (4−1)/(3−1)=3/2(4-1)/(3-1) = 3/2. BC: (4−1)/(6−4)=3/2(4-1)/(6-4) = 3/2. Both pairs parallel → parallelogram.
11Problem 11
Answer
(a) (3, 2) (b) See working
Full working
(a) Centroid = ((0+6+3)/3,(0+0+6)/3)=(3,2)((0+6+3)/3, (0+0+6)/3) = (3, 2).

(b) Medians from each vertex to the midpoint of the opposite side:
- From A to (4.5, 3): line y=(2/3)xy = (2/3)x. At (3, 2): y=2y = 2 ✓.
- From B to (1.5, 3): direction (-4.5, 3). Equation passes through (3, 2)?
- From C to (3, 0): vertical line x=3x = 3 passes through (3, 2) ✓.

All three medians intersect at the centroid (3, 2).
12Problem 12
Answer
(a) y=−2x−1y = -2x - 1 (b) y=−2x+1y = -2x + 1
Full working
(a) Reflection in xx-axis: y→−yy \to -y. So −y=2x+1⇒y=−2x−1-y = 2x + 1 \Rightarrow y = -2x - 1.

(b) Reflection in yy-axis: x→−xx \to -x. y=2(−x)+1=−2x+1y = 2(-x) + 1 = -2x + 1.