MTH 232

Section 8.3Connections 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 formula2.Midpoint formula3.Distance formula4.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.