StudentAidUnit 7: Coordinate Geometry — Review Exercise
9th Class Mathematics · Unit 7: Coordinate Geometry
R7.2.Find the distance between two points A(2, 3) and B(7, 8) on a coordinate plane.
Formula
Distance formula
d=sqrt{(x2-x1)2+(y2-y1)2}
Working
d=sqrt{(7-2)2+(8-3)2}=sqrt{25+25}
Result
d=sqrt{50}=5sqrt2
R7.3.Find the midpoint of the line segment joining the points (4, -2) and (-6, 3).
Formula
Midpoint formula
left(x1+x22,y1+y22right)
Working
left(-6+42,3-22right)=left(-22,frac12right)
Result
left(-1,frac12right)
R7.4.Calculate the gradient (slope) of the line passing through the points (1, 2) and (4, 6).
Formula
Slope formula
m=y2-y1x2-x1
Working
=6-24-1
Result
m=frac43
R7.5.Find the equation of the line in the form y = mx + c that passes through the points (3, 7) and (5, 11).
Formula
Slope
m=y2-y1x2-x1=11-75-3=frac42
m=2
Equation through (3,7)
y-7=2(x-3)
y-7=2x-6
Result
y=2x+1
R7.6.If two lines are parallel and one line has a gradient of 2/3, what is the gradient of the other line?
Given
m1=frac23
Formula
Parallel condition
m1=m2
Result
m2=frac23
R7.7.An airplane needs to fly from city A to coordinates (12, 5) to city B at coordinates (8, -4). Calculate the straight-line distance between these two cities.
Formula
Distance formula
d=sqrt{(x2-x1)2+(y2-y1)2}
Working
d=sqrt{(8-12)2+(-4-5)2}=sqrt{16+81}
Result
d=sqrt{97} ≈ 9.85 units
R7.8.In a landscaping project, the path starts at (2, 3) and ends at (10, 7). Find the midpoint.
Formula
Midpoint formula
left(x1+x22,y1+y22right)
Working
left(10+22,3+72right)=left(122,102right)
Result
(6,5)
R7.9.A drone is flying from point (2, 3) to point (10, 15) on the grid. Calculate the gradient of the line along which the drone is flying and the total distance travelled.
Formula
Gradient
m=y2-y1x2-x1=15-310-2=128
m=frac32
Distance
d=sqrt{(x2-x1)2+(y2-y1)2}
Working
d=sqrt{(10-2)2+(15-3)2}=sqrt{64+144}
Result
m=frac32, d=sqrt{208}=4sqrt{13} ≈ 14.4 units
R7.10.For a line with a gradient of -3 and a y-intercept of 2, write the equation of the line in: (a) Slope-intercept form (b) Point-slope form using the point (1, 2) (c) Two-point form using the points (1, 2) and (4, -7) (d) Intercepts form (e) Symmetric form (f) Normal form
Given
m=-3, c=2
Result
(a) Slope-intercept
y=-3x+2
(b) Point-slope, P(1,2)
y-2=-3(x-1)
(c) Two-point, (1,2) and (4,-7)
y-2-7-2=x-14-1 ⇒ y-2-9=x-13
(d) Intercept form
y=-3x+2 ⇒ y+3x=2 ⇒ y2+x2/3=1
(e) Symmetric/normal setup
y+3x=2; divide by sqrt{32+12}=sqrt{10}: frac{y}{sqrt{10}+frac{3x}{sqrt{10}=frac{2}{sqrt{10}
(f) Normal form
tanα=-3 ⇒ α=tan-1(-3)=-71.56^circ, p=frac{2}{sqrt{10}
(f) Result
xcos(-71.56^circ)+ysin(-71.56^circ)=frac{2}{sqrt{10}