Question

6 from unit 1, lesson 9 here are 3 points in the plane. explain how to determine whether point c is closer to point a or point b. 7 from unit 1, lesson 7 diego says a quadrilateral with 4 congruent sides is always a regular polygon. mai says it never is one. do you agree with either of them? explain your reasoning. learning targets + i can describe a reflection by specifying the line of reflection. + i can draw reflections.

Explanation:

Step1: Recall distance - formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ in a plane is given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Let the coordinates of point $A$ be $(x_A,y_A)$, point $B$ be $(x_B,y_B)$ and point $C$ be $(x_C,y_C)$.

Step2: Calculate distance between $C$ and $A$

$d_{CA}=\sqrt{(x_C - x_A)^2+(y_C - y_A)^2}$.

Step3: Calculate distance between $C$ and $B$

$d_{CB}=\sqrt{(x_C - x_B)^2+(y_C - y_B)^2}$.

Step4: Compare the distances

If $d_{CA}d_{CB}$, then point $C$ is closer to point $B$. If $d_{CA}=d_{CB}$, then point $C$ is equidistant from $A$ and $B$.

for question 7:
A regular polygon has all sides congruent and all angles congruent. A quadrilateral with 4 congruent sides is a rhombus. A rhombus does not always have all angles congruent (a square is a special case of a rhombus where all angles are right - angles). So, Diego is wrong because not all quadrilaterals with 4 congruent sides are regular polygons. Mai is also wrong because a square (a type of quadrilateral with 4 congruent sides) is a regular polygon.

Answer:

Calculate the distance between $C$ and $A$ using the distance - formula $d_{CA}=\sqrt{(x_C - x_A)^2+(y_C - y_A)^2}$, calculate the distance between $C$ and $B$ using the distance - formula $d_{CB}=\sqrt{(x_C - x_B)^2+(y_C - y_B)^2}$, and compare $d_{CA}$ and $d_{CB}$. If $d_{CA}d_{CB}$, $C$ is closer to $B$; if $d_{CA}=d_{CB}$, $C$ is equidistant from $A$ and $B$.