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 transformation type

Since it's a sequence of rigid - transformations (translations, rotations, reflections), the shape and size of the figure are preserved.

Step2: Use corresponding points

We know that the same sequence of transformations that takes \(A\) to \(A'\), \(B\) to \(B'\) and \(C\) to \(C'\) will take \(D\) to \(D'\). We can use the relationships between the known corresponding points (\(A - A'\), \(B - B'\), \(C - C'\)) to find \(D'\). For example, if it is a translation, we find the vector from \(A\) to \(A'\) and apply the same vector to \(D\). If it is a rotation about a certain point, we use the center of rotation and the angle of rotation to find the new position of \(D\). If it is a reflection, we use the line of reflection to find the image of \(D\).

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 \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Calculate distances

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

Step3: Compare distances

Compare the values of \(d_{AC}\) and \(d_{BC}\). 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 \(B\).

Step1: Recall the definition of a regular polygon

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

Step2: Analyze the quadrilateral

A quadrilateral with 4 congruent sides is a rhombus. In a rhombus, the angles are not always congruent (except when it is a square). So, a quadrilateral with 4 congruent sides is not always a regular polygon (since the angles may not be equal), but it can be a regular polygon (when it is a square). So, neither Diego nor Mai is correct.

Answer:

Draw the point \(D'\) such that the transformation from \(D\) to \(D'\) is consistent with the transformation from \(A\) to \(A'\), \(B\) to \(B'\) and \(C\) to \(C'\) based on the type of rigid - transformation (translation, rotation, reflection) and label it as \(D'\).

6.