To find when the formulas for the midpoint, [tex]\( x=\left(\frac{a}{a+b}\right)\left(x_2-x_1\right)+x_1 \)[/tex] and [tex]\( y=\left(\frac{a}{a+b}\right)\left(y_2-y_1\right)+y_1 \)[/tex], will simplify to the standard midpoint formula, consider the following steps:
1. Substitute the values given in each option into the condition for a midpoint:
Option 1: [tex]\(a = 1\)[/tex] and [tex]\(b = 2\)[/tex]:
- Compute [tex]\( a + b \)[/tex].
- If [tex]\( a + b = 1 + 2 = 3 \)[/tex], which does not simplify to the midpoint formula.
Option 2: [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex]:
- Compute [tex]\( a + b \)[/tex].
- If [tex]\( a + b = 2 + 1 = 3 \)[/tex], which does not simplify to the midpoint formula.
Option 3: [tex]\(a = 1\)[/tex] and [tex]\(a + b = 2\)[/tex]:
- If [tex]\(a + b = 2\)[/tex] and [tex]\( a = 1 \)[/tex],
- Then [tex]\( b = 2 - 1 = 1 \)[/tex].
- Thus, both [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are 1, which leads to [tex]\( a = 1 \)[/tex] and [tex]\( b = 1 \)[/tex], which simplifies to the symmetric form acknowledged by the midpoint formula [tex]\( x = \frac{x_1 + x_2}{2} \)[/tex] and [tex]\( y = \frac{y_1 + y_2}{2} \)[/tex].
Option 4: [tex]\(a = 2\)[/tex] and [tex]\(a + b = 2\)[/tex]:
- If [tex]\( a = 2 \)[/tex] and [tex]\( a + b = 2 \)[/tex],
- Then [tex]\( b = 2 - 2 = 0 \)[/tex], which cannot be correct as it does not relate to midpoint calculation simplification.
From analyzing each case:
The correct option is:
[tex]\( a = 1 \)[/tex] and [tex]\( a + b = 2 \)[/tex].
This corresponds to Option 3.
