Question

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b. guided construction

1. create a new blank desmos geometry file.
2. create two points, call them a and b. record their coordinates.
3. draw segment ab between the two points.
4. construct the mid - point m of segment ab. what are the coordinates of m?
5. construct the line perpendicular to ab passing through m. check: does the line split ab into two equal halves at 90°?
6. move points a or b to new positions. does the perpendicular bisector update correctly? why or why not?

c. practice problems

1. given two points (2,3) and (6,7), do the construction. record the mid - point and equation of the perpendicular line.
2. points (-4,1) and (2, - 5). do the construction. record the mid - point and perpendicular bisector.
3. choose your own two points. do the construction and share your work (screenshot: windows s shift)

d. reflection
a. what are the key tools used in this construction? list them in order.
b. why does the perpendicular bisector always cross the mid - point at right angles? explain in your own words.

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step2: Calculate mid - point for points $(2,3)$ and $(6,7)$

Let $(x_1,y_1)=(2,3)$ and $(x_2,y_2)=(6,7)$. Then $M_1=(\frac{2 + 6}{2},\frac{3+7}{2})=(4,5)$.

Step3: Calculate slope of line segment joining $(2,3)$ and $(6,7)$

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. So, $m_1=\frac{7 - 3}{6 - 2}=\frac{4}{4}=1$.

Step4: Find slope of perpendicular bisector

The slope of a line perpendicular to a line with slope $m$ is $m'=-\frac{1}{m}$. So the slope of the perpendicular bisector of the line segment joining $(2,3)$ and $(6,7)$ is $m'=- 1$.

Step5: Use point - slope form to find equation of perpendicular bisector

The point - slope form of a line is $y - y_0=m'(x - x_0)$, using the mid - point $(4,5)$ and $m'=-1$, we get $y - 5=-1(x - 4)$, which simplifies to $y=-x + 9$.

Step6: Repeat for points $(-4,1)$ and $(2,-5)$

Mid - point: $M_2=(\frac{-4 + 2}{2},\frac{1+( - 5)}{2})=(-1,-2)$.
Slope of line segment: $m_2=\frac{-5 - 1}{2-( - 4)}=\frac{-6}{6}=-1$.
Slope of perpendicular bisector: $m_2' = 1$.
Equation of perpendicular bisector using point - slope form with $(x_0,y_0)=(-1,-2)$ is $y+2 = 1\times(x + 1)$, or $y=x - 1$.

Step7: General property of perpendicular bisector

By definition, a perpendicular bisector of a line segment is a line that is perpendicular to the segment and passes through its mid - point. So it always crosses the mid - point at right angles. The key tools used in the construction are the mid - point formula and the relationship between slopes of perpendicular lines.

Answer:

For points $(2,3)$ and $(6,7)$: Mid - point is $(4,5)$, equation of perpendicular bisector is $y=-x + 9$.
For points $(-4,1)$ and $(2,-5)$: Mid - point is $(-1,-2)$, equation of perpendicular bisector is $y=x - 1$.
a. Key tools: Mid - point formula, slope relationship for perpendicular lines.
b. Yes, a perpendicular bisector always crosses the mid - point at right angles because that is its definition: a line that is perpendicular to a line segment and passes through its mid - point.