To find which combinations of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are true when finding the midpoint using the given formulas, we need to examine each option carefully.
The formulas for the midpoint of a directed line segment are:
[tex]\[
x = \left(\frac{a}{a+b}\right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left(\frac{a}{a+b}\right)(y_2 - y_1) + y_1
\][/tex]
We need to evaluate the given options to see which one results in a valid midpoint calculation:
1. [tex]\(a = 1\)[/tex] and [tex]\(b = 2\)[/tex]
[tex]\[
a + b = 1 + 2 = 3
\][/tex]
The midpoint formulas would then involve [tex]\(\frac{1}{3}\)[/tex], which is not directly indicative here.
2. [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex]
[tex]\[
a + b = 2 + 1 = 3
\][/tex]
The midpoint formulas would involve [tex]\(\frac{2}{3}\)[/tex], again not directly indicative here.
3. [tex]\(a = 1\)[/tex] and [tex]\(a + b = 2\)[/tex]
[tex]\[
b = 2 - 1 = 1
\][/tex]
This setup means [tex]\(a\)[/tex] is 1 and [tex]\(b\)[/tex] is 1 which adds up to 2.
4. [tex]\(a = 2\)[/tex] and [tex]\(a + b = 2\)[/tex]
[tex]\[
b = 2 - 2 = 0
\][/tex]
This implies b is zero which isn't typically useful for finding a midpoint in traditional sense as it nullifies part of the value.
For the given question, the combination [tex]\(a = 1\)[/tex] and [tex]\(a + b = 2\)[/tex] holds true when finding the midpoint as it meaningfully sets up the fractions and thus calculations within valid bounds. Only the third option ultimately fits logically and mathematically for midpoint determination.
Therefore, the correct combination is:
[tex]\[
a = 1 \quad \text{and} \quad a + b = 2
\][/tex]
This setup is valid and results in true for the given midpoint formula.
