Question

1. there is a sequence of rigid transformations that takes a to a, b to b, and c to c. the same sequence takes d to d. draw and label d: (from unit 1, lesson 10.) 6. here are 3 points in the plane. explain how to determine whether point c is closer to point a or point b. (from unit 1, lesson 9.) 7. diego says a quadrilateral with 4 congruent sides is always a regular polygon. mai say it never is one. do you agree with either of them? (from unit 1, lesson 7.)

Explanation:

5.

Step1: Identify the type of rigid - transformation

Since it's a sequence of rigid transformations (translations, rotations, reflections), the shape and size of the figure are preserved. We can use the relationships between the pre - image and image points of \(A\), \(B\), and \(C\) to find \(D'\).

Step2: Use vector or coordinate - based approach (if coordinates are given, assume vector approach here)

Find the vector from \(A\) to \(A'\), from \(B\) to \(B'\), and from \(C\) to \(C'\). Since the transformation is the same for all points, apply the same vector to point \(D\) to get \(D'\). Let \(\overrightarrow{AA'}=\vec{v}\), \(\overrightarrow{BB'}=\vec{v}\), \(\overrightarrow{CC'}=\vec{v}\), then \(\overrightarrow{DD'}=\vec{v}\). Starting from point \(D\), move along the vector \(\vec{v}\) to draw and label \(D'\).

6.

Step1: Recall the 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 \(A=(x_A,y_A)\), \(B=(x_B,y_B)\) and \(C=(x_C,y_C)\).

Step2: Calculate the distances \(d_{AC}\) and \(d_{BC}\)

\(d_{AC}=\sqrt{(x_C - x_A)^2+(y_C - y_A)^2}\) and \(d_{BC}=\sqrt{(x_C - x_B)^2+(y_C - y_B)^2}\).

Step3: Compare the distances

If \(d_{AC}d_{BC}\), then point \(C\) is closer to point \(B\). If \(d_{AC}=d_{BC}\), then point \(C\) is equidistant from \(A\) and \(B\).

7.

Step1: Recall the definition of a regular polygon

A regular polygon has all sides congruent and all angles congruent.

Step2: Analyze the statements

A quadrilateral with 4 congruent sides is a rhombus. In a rhombus, the angles are not necessarily congruent (except for the special case of a square, which is also a rhombus). Diego says it's always a regular polygon, which is wrong because non - square rhombuses are not regular. Mai says it never is one, which is also wrong because a square (a type of quadrilateral with 4 congruent sides) is a regular polygon. So, we agree with neither of them.

Answer:

We agree with neither Diego nor Mai.