Question

1. in the diagram below, $overline{ab}$ and $overline{de}$ bisect each other. if $f$ is the mid - point of $overline{cd}$, $ab = 20$, and $de = 12$, then find the value of $fc+cb$. show how you arrived at your answer.

reasoning

1. triangle $efg$ is shown. using a ruler marked in centimeters:

(a) locate and mark the midpoints of each side of triangle $efg$.
(b) connect the midpoints using line segments.
(c) measure the length of each of the line segments you drew in (b) and label on the diagram.
(d) what do you notice about these side lengths?

1. in the diagram below, line $r$ is the perpendicular bisector of $overline{mn}$.

(a) what special point is point $p$ on $overline{mn}$. explain how you know this.
(b) choose any other point on line $r$ and mark it. label it as point $q$.
(c) draw in $overline{mq}$ and $overline{nq}$. measure each of their lengths to the nearest tenth of a centimeter.
(d) what do you notice about point $q$ relative to points $m$ and $n$?
n - gen math geometry - unit 1 - beginning concepts - lesson 8
emathinstruction, red hook, ny 12571, © 2023

Explanation:

Problem 6

Step1: Use the property of bisecting lines

Since $\overline{AB}$ and $\overline{DE}$ bisect each other, we can use the concept of congruent - triangles or the mid - point properties. But we don't have enough information from the description about the relationship between $AB$, $DE$ and the segments $FC$ and $CB$ directly from this property. However, if we assume that there is some hidden congruence or parallel - line relationship (not given explicitly in the text but a common geometric setup), and since $F$ is the mid - point of $\overline{CD}$, we know that $FC=\frac{1}{2}CD$.
There is not enough information in the problem statement to solve for $FC + CB$ with the given $AB = 20$ and $DE=12$ values. We need more information such as the relationship between the lines $AB$ and $DE$ and the segments $CD$, $CB$ (e.g., are there congruent triangles formed by these lines and segments?). So, we cannot solve this problem with the given information.

Problem 7

Step1: Locate mid - points

To locate the mid - points of each side of $\triangle EFG$:
For side $\overline{EF}$, measure the length of $\overline{EF}$ with a ruler. Divide the length by 2 and mark the mid - point. Let's call it $M_1$.
For side $\overline{FG}$, measure the length of $\overline{FG}$ with a ruler. Divide the length by 2 and mark the mid - point. Let's call it $M_2$.
For side $\overline{EG}$, measure the length of $\overline{EG}$ with a ruler. Divide the length by 2 and mark the mid - point. Let's call it $M_3$.

Step2: Connect mid - points

Use a straight - edge to connect $M_1$, $M_2$ and $M_3$ with line segments.

Step3: Measure line segments

Measure the lengths of the three line segments formed by connecting the mid - points: $\overline{M_1M_2}$, $\overline{M_2M_3}$ and $\overline{M_3M_1}$ using the ruler and label them on the diagram.

Step4: Observe side lengths

We will notice that the lengths of the line segments connecting the mid - points of the sides of a triangle are half the lengths of the opposite sides of the original triangle. This is a well - known property of the mid - segment theorem in geometry, which states that the segment joining the mid - points of two sides of a triangle is parallel to the third side and half its length.

Problem 8

Step1: Identify special point

(a) Since line $r$ is the perpendicular bisector of $\overline{MN}$, point $P$ is the mid - point of $\overline{MN}$. We know this because the perpendicular bisector of a line segment intersects the line segment at its mid - point.

Step2: Choose a point on $r$

(b) Choose any point $Q$ on line $r$.

Step3: Draw line segments

(c) Draw $\overline{MQ}$ and $\overline{NQ}$. Then measure the lengths of $\overline{MQ}$ and $\overline{NQ}$ to the nearest tenth of a centimeter.

Step4: Observe the relationship

(d) We will notice that $MQ = NQ$. This is because any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment. This is a fundamental property of the perpendicular bisector.

Answer:

Step1: Identify special point

(a) Since line $r$ is the perpendicular bisector of $\overline{MN}$, point $P$ is the mid - point of $\overline{MN}$. We know this because the perpendicular bisector of a line segment intersects the line segment at its mid - point.

Step2: Choose a point on $r$

(b) Choose any point $Q$ on line $r$.

Step3: Draw line segments

(c) Draw $\overline{MQ}$ and $\overline{NQ}$. Then measure the lengths of $\overline{MQ}$ and $\overline{NQ}$ to the nearest tenth of a centimeter.

Step4: Observe the relationship

(d) We will notice that $MQ = NQ$. This is because any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment. This is a fundamental property of the perpendicular bisector.