Question

find the midpoint of points a(10,1) and b(4,−5) graphically. you may press submit or click the graph again to replot the midpoint.

Explanation:

Step1: Recall the midpoint formula

The midpoint \( M(x_m, y_m) \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( x_m=\frac{x_1 + x_2}{2} \) and \( y_m=\frac{y_1 + y_2}{2} \). For points \( A(10,1) \) and \( B(4, - 5) \), we identify \( x_1 = 10,y_1=1,x_2 = 4,y_2=-5 \).

Step2: Calculate the x - coordinate of the midpoint

Substitute \( x_1 = 10 \) and \( x_2 = 4 \) into the formula for \( x_m \): \( x_m=\frac{10 + 4}{2}=\frac{14}{2}=7 \).

Step3: Calculate the y - coordinate of the midpoint

Substitute \( y_1 = 1 \) and \( y_2=-5 \) into the formula for \( y_m \): \( y_m=\frac{1+(-5)}{2}=\frac{1 - 5}{2}=\frac{-4}{2}=-2 \).

Graphically, to find the midpoint, we can also:

• For the x - coordinates: The distance between \( x = 10 \) and \( x = 4 \) is \( 10-4 = 6 \). Half of this distance is \( \frac{6}{2}=3 \). Moving 3 units from \( x = 4 \) towards \( x = 10 \) (or 3 units from \( x = 10 \) towards \( x = 4 \)) gives \( 4 + 3=7 \) (or \( 10-3 = 7 \)).
• For the y - coordinates: The distance between \( y = 1 \) and \( y=-5 \) is \( 1-(-5)=6 \). Half of this distance is \( \frac{6}{2}=3 \). Moving 3 units from \( y=-5 \) towards \( y = 1 \) (or 3 units from \( y = 1 \) towards \( y=-5 \)) gives \( - 5+3=-2 \) (or \( 1 - 3=-2 \)).

Answer:

The midpoint of points \( A(10,1) \) and \( B(4,-5) \) is \( (7,-2) \).