Question

using the midpoint and distance formulas, calculate the coordinate of the midpoint and the length of the segment.
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \quad m = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \
ight)
graph with points a(1,2), d(3,3), c(3,1), b(5,2)
length of segment \overline{ab} \quad coordinates of midpoint
a. \sqrt{5}
b. 4
c. (3.5, 3)
d. (3, 2)

Explanation:

Step1: Identify coordinates of A and B

From the graph, \( A(1, 2) \) and \( B(5, 2) \).

Step2: Calculate length of \( \overline{AB} \)

Use distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Here, \( x_1 = 1, y_1 = 2, x_2 = 5, y_2 = 2 \).
Substitute: \( d = \sqrt{(5 - 1)^2 + (2 - 2)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4 \). Wait, but option a is \( \sqrt{5} \), maybe I misread coordinates. Wait, maybe A is (1,2) and B is (3,3)? No, the graph shows A(1,2), B(5,2)? Wait, no, maybe the points are A(1,2) and B(3,3)? Wait, no, let's check again. Wait, the options for length: a. \( \sqrt{5} \), b. 4. Wait, maybe A(1,2) and B(2,3)? No, the graph has A(1,2), D(3,3), C(3,1), B(5,2). So A(1,2), B(5,2): distance is 4 (option b). Midpoint: \( M = (\frac{1 + 5}{2}, \frac{2 + 2}{2}) = (3, 2) \) (option d). Wait, but let's recheck. Wait, maybe the points are A(1,2) and B(2,3)? No, the coordinates given: A(1,2), B(5,2). So distance: \( \sqrt{(5 - 1)^2 + (2 - 2)^2} = 4 \), midpoint \( (3, 2) \). But wait, if A is (1,2) and B is (2,3), distance would be \( \sqrt{(2 - 1)^2 + (3 - 2)^2} = \sqrt{2} \), not matching. Wait, the graph: A is at (1,2), B at (5,2). So:

Length of \( \overline{AB} \):
\( d = \sqrt{(5 - 1)^2 + (2 - 2)^2} = \sqrt{16 + 0} = 4 \) (option b).

Midpoint of \( \overline{AB} \):
\( M = (\frac{1 + 5}{2}, \frac{2 + 2}{2}) = (3, 2) \) (option d).

Wait, but the options for length: a. \( \sqrt{5} \), b. 4. For midpoint: c. (3.5,3), d. (3,2). So if A(1,2) and B(5,2), then length is 4 (b), midpoint (3,2) (d).

Answer:

Length of Segment \( \overline{AB} \): b. 4
Coordinates of Midpoint: d. (3, 2)