MPM2D     Date:  ________________________________  

2.1 Midpoint of a Line Segment

Learning Goals Success Criteria Develop a formula for the midpoint of a line segment.

o Understand the meaning of the midpoint o Know the formula to solve for midpoint

Use this formula to solve problems. o Solve for the midpoint of a line segment o Determine the median of a triangle o Determine the right bisector of a triangle

From MIDPOINT INVESTIGATION, we have that the midpoint of a line segment with endpoints 𝐴 𝑥!,𝑦! and 𝐵 𝑥!,𝑦! is

𝑀𝑥! + 𝑥!2 ,

𝑦! + 𝑦!2

i.e. the x-value is the average of the endpoints’ x-values and

the y-value is the average of the endpoints’ y-values. Example 1: A city has two hospitals shown on the city map at coordinates A(3, 5) and B(11, 14). The city wants to build a new ambulance station halfway between the two hospitals. Determine the coordinates of this location.      

𝑥, 𝑦 =   !!!!!!

, !!!!!!

= !!!!

!, !!!"

!

= !"!, !"!

= 7, 9.5 The coordinates of the new ambulance station are 7, 9.5 .

0 2 4 6 8 10 12 14

2

4

6

8

10

12

14

Example 2: Determine the equation for the median from vertex R for the triangle with vertices P(-5, 1), Q(1, 5), and R(3, -3). The equation for any line is 𝑦 = 𝑚𝑥 + 𝑏 So we need to find 𝑚 (slope) and 𝑏 (y-intercept). Step 1: Solve for coordinates of T 𝑇 𝑥,𝑦 =   !!!!!

!, !!!!!

!

= !!!!!, !!!!

= − !

!, !!

= −2, 3 Step 2: Use points R and T to solve for slope. 𝑚 = !"!"

!"#

= !!!!!! !!

   

  = !!!    

Step 3: Use either point in 𝑦 = 𝑚𝑥 + 𝑏  to solve for 𝑏 𝑦 = 𝑚𝑥 + 𝑏 3 = − !

!−2 + 𝑏

3 = !"!+ 𝑏

!"!− !"

!= 𝑏

!!= 𝑏

Therefore the equation of the median is 𝑦 = − !!𝑥 + !

!

Example 3: Two schools are located at points C(-4, 3) and D(2, -5) on a top map. The school board would like to build a new sports complex at a location that is equidistant from both schools. Determine an equation to represent all possible locations for the sports complex. Note: The sports complex does not have to be directly between the two schools. It can be any point on the right bisector of the line segment. The equation for any line is 𝑦 = 𝑚𝑥 + 𝑏 So we need to find 𝑚 (slope) and 𝑏 (y-intercept) for the right bisector. Step 1: Midpoint 𝑥,𝑦 =   !!!!!

!, !!!!!

!

= !!!!!, !!!!

= !!

!, !!!

= −1,−1 Step 2: Slope 𝑚 = !"#$

!"#

= !!!!!! !!

   

  = !!!  

  = !!!    

VERY IMPORTANT: the right bisector is perpendicular to CD. The slope of the right bisector is the negative reciprocal.

𝑚! =34

Step 3: Find 𝑏

𝑦 = 𝑚𝑥 + 𝑏 −1 =

34 −1 + 𝑏

−1+34 = 𝑏

−44+

34 = 𝑏

−14 = 𝑏

The equation 𝑦 = !!𝑥 − !

! represents all possible locations for the sports complex.