Math  /  Geometry

QuestionFor the graph shown to the right, find (a) ABA B to the nearest tenth and (b) the coordinates of the midpoint of AB\overline{A B}. a. AB=A B= \square (Round to the nearest tenth as needed.)

Studdy Solution

STEP 1

1. The coordinates of point A A are (4,2) (4, 2) .
2. The coordinates of point B B are (4,2) (-4, -2) .
3. We will use the distance formula to find AB AB .
4. We will use the midpoint formula to find the midpoint of AB\overline{AB}.

STEP 2

1. Calculate the distance AB AB .
2. Find the coordinates of the midpoint of AB\overline{AB}.

STEP 3

To find the distance AB AB , use the distance formula: AB=(x2x1)2+(y2y1)2 AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

STEP 4

Substitute the coordinates of points A(4,2) A(4, 2) and B(4,2) B(-4, -2) into the distance formula: AB=((4)4)2+((2)2)2 AB = \sqrt{((-4) - 4)^2 + ((-2) - 2)^2}

STEP 5

Calculate the differences: x2x1=44=8 x_2 - x_1 = -4 - 4 = -8 y2y1=22=4 y_2 - y_1 = -2 - 2 = -4

STEP 6

Substitute the differences into the formula: AB=(8)2+(4)2 AB = \sqrt{(-8)^2 + (-4)^2} AB=64+16 AB = \sqrt{64 + 16} AB=80 AB = \sqrt{80}

STEP 7

Calculate the square root and round to the nearest tenth: AB808.9 AB \approx \sqrt{80} \approx 8.9

STEP 8

To find the midpoint of AB\overline{AB}, use the midpoint formula: M=(x1+x22,y1+y22) M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

STEP 9

Substitute the coordinates of points A(4,2) A(4, 2) and B(4,2) B(-4, -2) into the midpoint formula: M=(4+(4)2,2+(2)2) M = \left( \frac{4 + (-4)}{2}, \frac{2 + (-2)}{2} \right)

STEP 10

Calculate the coordinates of the midpoint: M=(02,02) M = \left( \frac{0}{2}, \frac{0}{2} \right) M=(0,0) M = (0, 0)
The distance AB AB is approximately 8.9 \boxed{8.9} and the coordinates of the midpoint of AB\overline{AB} are (0,0) \boxed{(0, 0)} .

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