Select the correct answer.

The endpoints of [tex]\overline{GH}[/tex] are [tex]G(10,1)[/tex] and [tex]H(3,5)[/tex]. What is the midpoint of [tex]\overline{GH}[/tex]?

A. [tex](-4,9)[/tex]
B. [tex]\left(\frac{7}{2}, 2\right)[/tex]
C. [tex]\left(\frac{13}{2}, 3\right)[/tex]
D. [tex](13,6)[/tex]



Answer :

Solution:

To find the midpoint of a line segment with endpoints [tex]\( G \)[/tex] and [tex]\( H \)[/tex], we use the midpoint formula. The midpoint formula for the endpoints [tex]\( G (x_1, y_1) \)[/tex] and [tex]\( H (x_2, y_2) \)[/tex] is:

[tex]\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]

Given the endpoints [tex]\( G (10, 1) \)[/tex] and [tex]\( H (3, 5) \)[/tex]:

1. Identify the coordinates of [tex]\( G \)[/tex] and [tex]\( H \)[/tex]:
- [tex]\( G (10, 1) \)[/tex]: Where [tex]\( x_1 = 10 \)[/tex] and [tex]\( y_1 = 1 \)[/tex]
- [tex]\( H (3, 5) \)[/tex]: Where [tex]\( x_2 = 3 \)[/tex] and [tex]\( y_2 = 5 \)[/tex]

2. Apply the midpoint formula:
[tex]\[ \left( \frac{10 + 3}{2}, \frac{1 + 5}{2} \right) \][/tex]

3. Perform the calculations:
- For the x-coordinate:
[tex]\[ \frac{10 + 3}{2} = \frac{13}{2} = 6.5 \][/tex]

- For the y-coordinate:
[tex]\[ \frac{1 + 5}{2} = \frac{6}{2} = 3.0 \][/tex]

4. Combine the results to get the midpoint:
[tex]\[ \left( 6.5, 3.0 \right) \][/tex]

So, the midpoint of [tex]\( \overline{ GH } \)[/tex] is [tex]\( \left( 6.5, 3.0 \right) \)[/tex].

Among the given choices, the correct answer matches option C:
[tex]\[ \boxed{\left( \frac{13}{2}, 3 \right)} \][/tex]