MPM2D1 – Unit 2, 3: Analytic Geometry and Geometric Properties Review
Length and Midpoint of a Line Segment
1.
i) Calculate the length of the line segment with the following pair of endpoints.
ii) Determine the coordinates of the midpoint of the line segment with each pair
of end points.
a) C (-3, -9), D (-5, 11)
b) M (16, 12), N (19, 8)
The Equation of a Line
2.
For each part, the coordinates of two points are given. Graph the line through
each pair of points. Determine the equation of the line. Write the equation in
slope y-intercept form. Check that the coordinates of the points satisfy the
equation.
a) H (4, 1), J (1, -2)
3.
Which of the given points lie on the line with equation 5x - 3y + 30 = 0?
i) A (-3, 5)
4.
ii) B (3, 15)
b) 2x - 5y + 7 = 0
d) 4x + 3y + 25 = 0
Determine the equation of each line. Write the equation in standard form.
a) slope 3, through A (5, 7)
6.
iii) C (2, -5)
Which of the lines below pass through the point C (-4, -3)?
a) x + 2y + 10 = 0
c) 3x - 4y + 1 = 0
5.
b) K (5, 13), L (-2, -1)
b) slope 4, x-intercept = –1
Line segment CD has endpoints C (-3, 2) and D (1, 8).
a) Determine the equation of the perpendicular bisector of CD.
b) Show that the perpendicular bisector passes through the point E (5, 1).
c) Verify that E is equidistant from C and D.
Distance Between a Point and a Line
7.
Determine the shortest distance between P(5, 1) and line 0 = x – 2y + 14.
The Equation of a Circle
8.
Determine if each point is on the circle defined by x2 + y2 = 49.
a) A (0, 7)
9.
b) B (3, 6)
A circle has centre O (0, 0) and radius 10. Which points lie on the circle?
i) E (-10, 0)
ii) F (6, -8)
iii) G (3, -9)
Investigations
10.
Points A (-4, 6), B (16, 14), and C (20, 4) are vertices of triangle ABC.
a) Verify that triangle ABC is a right angle.
b) M is the midpoint of AC. Verify that MB = MA = MC.
11.
a) For the point F (6, 10). Calculate the length of the line segment joining F to
each point.
i) A (1, 10)
ii) B (1, 14)
iii) C (1, 18)
b) The line segment joining E (1, a) to F has the same length as CF.
Determine the two possible values of a.
Properties of Triangles and Quadrilaterals
12.
Draw a triangle with vertices J(8, 8), K(-5, -5), and L(5, -7). What type of triangle
does ΔJKL appear to be?
a) Use analytic geometry to verify your classification of ΔJKL.
b) Determine the perimeter of the triangle.
c) Determine the area of the triangle.
13.
A triangle has vertices P(a, b), Q(c, d), and R(e, f).
a) Determine the coordinates of the midpoints of the sides.
b) Verify that ST is parallel to QR.
14.
Draw a quadrilateral with vertices P(-3, -1), Q(3, 1), R(7, 5) and S(1, 3).
a) Verify that the diagonals bisect each other.
b) Classify the quadrilateral PQRS. Justify your answer.