MTH 232
Section 8.3
Connections Between Algebra &
Geometry
Key Strategy
• In solving geometric problems, it is sometimes
easier and faster to place the figures in a
Cartesian plane and use algebra.
• Formulas needed from the previous section:
1. Slope formula
2. Midpoint formula
3. Distance formula
4. Pythagorean Theorem
Triangles
• Equilateral Triangles: use the distance formula
to find the distance between pairs of points.
These distances will be the lengths of the
sides of the triangle. If all three lengths are
the same, the triangle is equliateral.
• Isosceles Triangles: same idea, except only
two of the three lengths need to be the same.
• Right Triangles: show that the lengths satisfy
the Pythagorean Theorem.
Parallel and Perpendicular Lines
• Two lines in the plane are parallel if and only if
their slopes are equal.
• Two lines in the plane are perpendicular if and
only if the product of their slopes is -1
(another way to state this is to say that the
slopes are opposite reciprocals).
Examples
• Determine if the triangle formed by A(-3,3),
B(1,-2), and C(6,2) is an isosceles right
triangle.
• Find the equation of the line that passes
through (7, -4) and is parallel to the line given
by y = 5x + 8.
More Definitions
• An altitude of a triangle is a line through the
vertex of a triangle that is perpendicular to
the opposite side.
• A perpendicular bisector of a line segment is a
line that passes through the midpoint of the
line segment at a 90-degree angle.
More Examples
• Find the equation of the perpendicular
bisector of the line segment with endpoints
P(3,-1) and Q(-5,7).
• Find the equation of the altitude through
point B of the previous triangle example. Does
that altitude necessarily pass through the
midpoint of side AC? Explain.