Use matrices to determine the coordinates of the vertices of the reflected figure. then graph the pre-image and the image on the same coordinate grid.

The coordinates of the vertices of the reflected figure are :

R' is (-3 , -7), S' is (5 , -3), T' is (6 , 5) ⇒the right answer is figure (a)

Step-by-step explanation:

* Lets study the matrices of the reflection

- The matrix of the reflection across the x-axis is

- Because when we reflect any point across the x-axis we

 change the sign of the y-coordinate

- The matrix of the reflection across the y-axis is

- Because when we reflect any point across the y-axis we

 change the sign of the x-coordinate

* Now lets solve the problem

- We will multiply the matrix of the reflection across the x-axis

 by each point to find the image of each point

- The dimension of the matrix of the reflection across the x-axis

 is 2×2 and the dimension of the matrix of each point is 2×1,

then the dimension of the matrix of each image is 2×1

∵ Point R is (-3 , 7)

∴

∴ R' is (-3 , -7)

∵ Point S is (5 , 3)

∴

∴ S' is (5 , -3)

∵ Point T is (6 , -5)

∴

∴ T' is (6 , 5)

* Look to the answer and find the correct figure

- In figure (d) R' is (-3 , -7), S' is (5 , -3), T' is (6 , 5)

∴ The right answer is figure (a)

answer: 590

step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240