Question

1. the coordinates of a triangle are given in the table. the triangle is translated using the rule $(x,y)\to(x - 3,y + 2)$. what are the coordinates of the image of the triangle?

1. translate image 3 units left and 4 units down.

a. explain why the area of rectangle a b c d is congruent to the area of rectangle $abcd$?

b. is angle a and angle $a$ congruent? explain.

Explanation:

3. Step1: Apply the translation rule to each coordinate

For the point \((0,1)\):
\(x = 0-3=-3\), \(y = 1 + 2=3\). So the new coordinate is \((-3,3)\)

Step2: Apply the translation rule to the second coordinate

For the point \((2,3)\):
\(x=2 - 3=-1\), \(y = 3+2 = 5\). So the new coordinate is \((-1,5)\)

Step3: Apply the translation rule to the third coordinate

For the point \((4,2)\):
\(x=4-3 = 1\), \(y=2 + 2=4\). So the new coordinate is \((1,4)\)

4. A. Step1: Recall the property of translation

Translation is a rigid transformation. Rigid transformations (such as translation) preserve the shape and size of a figure.
The area of a rectangle is calculated by \(A = l\times w\). Since translation does not change the length and width of the rectangle \(ABCD\) (because it is a rigid transformation), the area of rectangle \(A'B'C'D'\) (the translated image) is equal to the area of rectangle \(ABCD\).

B. Step1: Recall the property of translation regarding angles

Translation is a rigid transformation. Rigid transformations preserve the measures of angles.
Since angle \(A\) in rectangle \(ABCD\) and angle \(A'\) in rectangle \(A'B'C'D'\) are related by a translation (a rigid transformation), \(m\angle A=m\angle A'\). So angle \(A\) and angle \(A'\) are congruent.

Answer:

3.
Image coordinates: \((-3,3)\), \((-1,5)\), \((1,4)\)
4.
A. Translation is a rigid transformation. Rigid - transformations preserve the shape and size of a figure. Since the area of a rectangle \(A = l\times w\) and translation does not change \(l\) and \(w\), the areas are congruent.
B. Yes. Translation is a rigid transformation. Rigid transformations preserve the measures of angles. So \(m\angle A=m\angle A'\), and \(\angle A\cong\angle A'\)