The coordinates of the vertices of triangle ABC are A(1,6), B(2,9), and C(7,10). Identify A'''B'''C''' after the following transformations; reflection y=x, reflection x-axis, and translation (x -7, y + 4) respectively.

A'''(-1,3), B'''(2,2) and C'''(3,-3)

Step-by-step explanation:

Hello There!

So the first thing we need to do is reflect the coordinates given over the y = x line which would be a straight line across the origin with a rate of change of 1

The rule for a reflection over the y = x line is  (x,y) > (y,x)

so pretty much we're just flipping the y and x values

(1,6) ---> (6,1)

(2,9) ---> (9,2)

(7,10) ---> (10,7)

so the new coordinates of A'B'C' are A'(6,1) , B'(9,2), and C'(10,7)

Now we want to reflect the points over the x axis

The rule for reflection over the x axis is (x,y) > (x,-y)

so the x value stays the same and the y values sign just get flipped so if the y value was originally negative then it will change to positive vise versa

(6,1) ---> (6,-1)

(9,2) ---> (9,-2)

(10,7) ---> (10,-7)

So the coordinates of A''B''C'' are  A''(6,-1) , B''(9,-2), and C''(10,-7)

Our final step is to translate 7 units to the left 4 units up

so the rule would be (x,y) > (x - 7, y + 4)

6-7=-1 | -1+4=3

(6,-1) ---> (-1,3)

9-7=2 | -2+4=2

(9,-2) ---> (2,2)

10-7=3 | -7+4=-3

(10,-7) ---> (3,-3)

so the after the three transformations the coordinates would be

A'''(-1,3), B'''(2,2) and C'''(3,-3)

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2

the arc length is found by finding the circumference of the full circle (2xpixradius) and then dividing by 4 to find just one quarter of the circle (90 degrees).