Download Unit 8 Day 1 Notes - Garnet Valley School District

Algebra II/Trigonometry Honors
Unit 8 Day 1: Apply the Distance and Midpoint Formulas
Objective: To find the length and midpoint of a line segment
To find the distance between two points, we will use
a formula similar to Pythagorean Theorem for right triangles.
The Distance Formula
The distance d between x1 , y1  and x2 , y2  is ___________________________________________
Example 1: Use the distance formula
Find the distance between (-3, 5) and (4, -1).
Example 2: Classify a triangle using the distance formula
The vertices of a triangle are A(4, 6), B(7, 3) and C(2, 1) as shown to the right. Classify ∆ABC as
scalene, isosceles, or equilateral.
The Midpoint Formula
A line segment’s midpoint is equidistant from the segment’s endpoints.
The formula for the midpoint of the line segment joining A x1 , y1  and B x2 , y2  is:
Example 3: Find the midpoint of a line segment
Find the midpoint of the line segment joining (-5, 1) and (-1, 6).
Example 4: Find a perpendicular bisector
Write an equation for the perpendicular bisector of the line segment joining A(-3, 4) and B(5, 6).
Recall: Perpendicular lines have opposite reciprocal slopes.
Perpendicular bisectors cut a segment into two equal parts.
The point-slope form y  y1  mx  x1  can be used to write the equation of a line.
Finding a Circle’s Center
The perpendicular bisector of any chord on
a circle passes through the circle’s center.
Given three points, you can use this theorem
to find the center of a circle.
Example 5: Solve a multi-step problem
Many scientists believe that an asteroid slammed into Earth about 65 million years ago on what is now
Mexico’s Yucatan peninsula, creating an enormous crate that is now deeply buried by sediment. Use the
labeled points on the outline of the circular crater to estimate its diameter. (Each unit in the coordinate
plane represents 1 mile.)
Step 1: Write equations for the perpendicular
bisectors of AO and OB using the method
from the previous example.
Step 2: Find the coordinates of the center of
the circle, where AO and OB intersect.
Step 3: Calculate the radius (the distance
between the center and any of the given
points on the circle) using the distance
formula.
HW: Page 493 #3-30 (M3), 31-35 odd, 41, 43, 52