Question

name____ 1.3 midpoints and segment bisectors period____ vocabulary vocabulary word definition picture midpoint the point that divides a segment into two congruent segments b is a midpoint segment bisector a point, ray, line, line segment or plane that intersects a circle in the interior or exploration 1 1. find the midpoint of a line segment. graph line segment ab, where the points a and b are shown below. 2. explain how to bisect line segment ab; that is, divide ab into two congruent segments. then bisect ab and use the result to find the midpoint m of ab. 3. what are the coordinates of the midpoint m? 4. compare the x - coordinates of a, b, and m. how are the coordinates of the midpoint m related to the coordinates of a and b exploration 2 5. add point c to your graph at (3, - 2). 6. use the pythagorean theorem to find the length of line segment ab. 7. how else can you determine the length of ab? 8. use the pythagorean theorem and point m to find the lengths of line segments am and mb. what can you conclude? a(3,4) b(-5,-2)

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1=3,y_1 = 4,x_2=-5,y_2=-2$.

Step2: Calculate x - coordinate of mid - point

$x_M=\frac{3+( - 5)}{2}=\frac{3 - 5}{2}=\frac{-2}{2}=-1$.

Step3: Calculate y - coordinate of mid - point

$y_M=\frac{4+( - 2)}{2}=\frac{4 - 2}{2}=\frac{2}{2}=1$.

Step4: Use Pythagorean theorem for length of AB

The distance $d$ between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $A(3,4)$ and $B(-5,-2)$, we have $d_{AB}=\sqrt{(-5 - 3)^2+(-2 - 4)^2}=\sqrt{(-8)^2+(-6)^2}=\sqrt{64 + 36}=\sqrt{100}=10$.

Step5: Use Pythagorean theorem for AM and MB

For $A(3,4)$ and $M(-1,1)$: $d_{AM}=\sqrt{(-1 - 3)^2+(1 - 4)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$. For $M(-1,1)$ and $B(-5,-2)$: $d_{MB}=\sqrt{(-5+1)^2+(-2 - 1)^2}=\sqrt{(-4)^2+(-3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$. We can conclude that $AM = MB$ since $M$ is the mid - point of $AB$.

Answer:

1. The mid - point formula is used to find the mid - point of a line segment.
2. To bisect a line segment, we can use a compass and straightedge construction or the mid - point formula.
3. The coordinates of the mid - point $M$ of $A(3,4)$ and $B(-5,-2)$ are $(-1,1)$.
4. The $x$ - coordinate of the mid - point $M$ is the average of the $x$ - coordinates of $A$ and $B$, i.e., $x_M=\frac{x_A+x_B}{2}$, and the $y$ - coordinate of the mid - point $M$ is the average of the $y$ - coordinates of $A$ and $B$, i.e., $y_M=\frac{y_A + y_B}{2}$.
5. Point $C(3,-2)$ is added to the graph.
6. The length of line segment $AB$ using the Pythagorean theorem is $10$.
7. We can also use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the length of $AB$.
8. The lengths of $AM$ and $MB$ are both $5$, and we can conclude that $AM = MB$ because $M$ is the mid - point of $AB$.