Number, Relations & Functions 10Z
Coordinate Geometry Proofs
Proving a Triangle is a Right Triangle.
Right Triangle: _________________________________________________________________________
Method 1: Show two sides of the triangle are perpendicular by demonstrating their slopes are negative reciprocals.
Method 2: Calculate the distances of all three sides and then test the Pythagorean’s Theorem to show the three lenghts
make the Pythagorean’s Theorem true.
1. Prove triangle ABC, with vertices A (4, -1), B (5, 6) and C (1,3), is an isosceles right triangle.
Proving a Quadrilateral is a Parallelogram
Parallelogram: ________________________________________________________________________
Method 1: Show that the diagonals bisect each other by showing the midpoints of the diagonals are the same
Method 2: Show both pairs of opposite sides are parallel by showing they have equal slopes
Method 3: Show both pairs of opposite sides are equal by using distance.
Method 4: Show one pair of sides is both parallel and equal.
2. Prove quadrilateral RSTU, with vertices R (3,2), S (6,2), T (0,-2), and U (-3,-2), is a parallelogram.
Proving a Quadrilateral is a Rectangle
Rectangle: ____________________________________________________________________________
Prove it is a parallelogram first, then
Method 1: Show that the diagonals are congruent.
Method 2: Show that it has a right angle by using slope.
3. Prove a quadrilateral with vertices G (1, 1), H (5, 3), I (4, 5) and J (0, 3) is a rectangle.
Proving a Quadrilateral is a Rhombus
Rhombus: ____________________________________________________________________________
Prove it is a parallelogram first, then
Method 1: Prove that the diagonals are perpendicular.
Method 2: Prove that a pair of adjacent sides are equal.
Method 3: Prove that all four sides are equal.
4. Prove that the quadrilateral with the vertices A (-1, 4), B (2, 6), C (5, 4) and D (2, 2) is a rhombus.
Number, Relations & Functions 10Z
Proving a Quadrilateral is a Square
Square: _______________________________________________________________________________
There are many ways to do this. For example, prove the diagonals bisect each other (parallelogram), are equal
(rectangle) and are perpendicular (rhombus).
5. Prove that the quadrilateral with vertices A (2, 2), B (5, -2), C (9, 1) and D (6, 5) is a square.
Proving a Quadrilateral is a Trapezoid
Trapezoid : _____________________________________________________________________________
Show one pair of sides are parallel (same slope) and one pair of sides are not parallel (different slopes).
6. Prove a quadrilateral with vertices K (1, 5), A (4, 7), T (7, 3) and E (1, -1) is a trapezoid.
Proving a Quadrilateral is an Isosceles Trapezoid
Isosceles Trapezoid: _______________________________________________________________________
Prove it is a trapezoid first, then
Method 1: Prove the diagonals are congruent using distance.
Method 2: Prove that the pair of non-parallel sides are equal.
7. Prove that the quadrilateral MILK with vertices M (1, 3), I (-1, 1), L (-1, -2) and K (4, 3) is an isosceles
trapezoid.
Number, Relations & Functions 10Z
Calculating the Area of Polygons in Coordinate Geometry
1. Find the area of triangle ABC with vertices A (7, 6), B (-3, -2) and C (11, -4).
Number, Relations & Functions 10Z
Practice:
1. Determine the point C on the x-axis that is equidistance from A (-2,3) and B (1,-5). (Solution: 13/6)
2. Two of the vertices of ABC are B (-3, -2) and C (5, 4). The midpoint of AB is M (-3, 2). Find
a. the coordinates of vertex A
b. the area of ABC
c. the coordinates of N, the midpoint of AC
d. the area of AMN
e. the ratio of AM to AB
f. the ratio of the area of AMN to the area of ABC
3. Find the vertices of the triangle whose sides lie on the lines x -3y + 8 = 0, 7x – y – 24 = 0, x + y = 0.
4. The vertices of ABC are A (5, 6), B (1, 2) and C (9, 4). Show that the line segment joining the midpoint of AB
to the midpoint of AC is
a. Parallel to BC
b. One –half the length of BC
5. Find the fourth vertex of a parallelogram with vertices (3, 5), (-2, 1) and (6, 2). (There are three answers, find
one).
6. Quadrilateral ABCD has vertices A (5, 3), B (-3, 7), C (-5, -1) and D (3, -5). Show that the lines joining the
midpoints of the opposite sides of the quadrilateral bisect each other.