Exercise 19 Page 401

Let's recall that the midpoint of a segment is the point that divides the segment into 2 congruent segments. To find the midpoint, we can use the Midpoint Formula.

The Midpoint Formula

The coordinates of the midpoint of a segment are the averages of the x-coordinates and of the y-coordinates

Let A( x_1, y_1) and B( x_2, y_2) be the endpoints of a segment AB. Then the midpoint of this segment has the following coordinates. (x_1+ x_2/2,y_1+ y_2/2) Now, we can use the Midpoint Formula to find the missing endpoint by splitting the usual formula into two separate equations — one for the x-coordinate and one for the y-coordinate. M(x_1+x_2/2, y_1+y_2/2) ⇕ x_M=x_1+x_2/2 and y_M=y_1+y_2/2 To find x_2, we can use the x-coordinate of the midpoint, x_M= 4, and the given endpoint, x_1= 5.

x_M=x_1+x_2/2

SubstituteII

x_M= 4, x_1= 5

4=5+x_2/2

MultEqn

LHS * 2=RHS* 2

8=5+x_2

SubEqn

LHS-5=RHS-5

3=x_2

RearrangeEqn

Rearrange equation

x_2=3

The x-coordinate of the endpoint H is 3. Now, we will determine its y-coordinate in a similar process. The y-coordinates of the midpoint and given endpoint are 3 and -6, respectively.

y_M=y_1+y_2/2

SubstituteII

y_M= 3, y_1= -6

3=-6+y_2/2

MultEqn

LHS * 2=RHS* 2

6=-6+y_2

AddEqn

LHS+6=RHS+6

12=y_2

RearrangeEqn

Rearrange equation

y_2=12

The y-coordinate of H is 12, so point H is located at (3,12).