Which transformations of quadrilateral PQRS would result in the image of the quadrilateral being located only in the first quadrant of the coordinate plane? Select each correct answer.

A. a translation right 4 units
B. a reflection across x = 4
C. a reflection across y = −x
D. a rotation of 90˚ counterclockwise about vertex Q
E. a reflection across x=3, then a translation up 8 units
F. a reflection across the x-axis, then a translation up 13 units

E

Step-by-step explanation:

The first quadrant is the top-right one

If a point is in the first quadrant, the x- and y-values of the point are both positive

Let's go through the one by one.

A) Translating the quadrilateral right 4 units will make the new S to be , which is not in the first quadrant.

B) Reflection the quadrilateral across will not change the y-value of any points. So, the new S will stay below the x-axis, and thus will never be in the first quadrant, because it will always have a negative y-value.

C) This one is tricky. We can prove(the proof is just taking cases of which quadrant) that if a point is in the second or fourth quadrant, it's reflection across the line will always be in the same quadrant it started in(i.e. reflect a point in the second quadrant, you get a point in the second quadrant). We know that P is in the second quadrant, so the new P will also be in the second quadrant. Therefore, the new quadrilateral will not be entirely in the first quadrant.

D) We intuitively(and can prove) that because R is right underneath Q, the new R will be directly to the left of Q, with the same distance as Q to original R. So, the new R will be using above method. This point is not in the first quadrant.

E) We know intuitively(and can prove) that a reflection across the line will make the entire quadrilateral in the first and fourth quadrant. However, When we translate up, the new S, the lowest point on the quadrilateral, will be above the x-axis, making all the points in the first quadrant! Therefore, E works.

F) When we reflect across the x-axis, any points in the second quadrant go to the third quadrant. So, the new P will be in the third quadrant. However, when we translate up 13 units, the new P will go to the second quadrant. It will never cross the y-axis, and will always have a negative x-coordinate. Therefore, P will not be in the first quadrant.

Summing up the cases that work, we get that is the only answer.

The number of orchestra seats=100

The number of main seats=240

The number of balcony seats=160

Step-by-step explanation:

Total number of seats in the theater=500

Let the number of orchestra seats=x

Let the number of main seats=y

Let the number of balcony seats=z

x+y+z=500

Cost of Orchestra Seats =$60

Cost of Main Seats =$45

Cost of Balcony Seats =$30

If all the seats are sold, the gross revenue to the theater is $ 21,600.

60x+45y+30z=21,600.

If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $18,600.

45y+30z+x(60)=18,600

We solve the three equations simultaneously

x+y+z=500

60x+45y+30z=21,600.

30x+45y+30z=18,600

Subtract the third equation from the second

30x=3000

x=100

Substitute x=100 into the equations x+y+z=500

100+y+z=500

y+z=500-100

y=400-z

Substitute y=400-z and x=100 into 30x+45y+30z=18,600

30x+45y+30z=18,600

30(100)+45(400-z)+30z=18600

3000+18000-45z+30z=18600

-15z=18600-21000

-15z=-3600

z=240

y=400-z=400-240=160

Therefore: x=100, y=240, z=160

The number of orchestra seats=100

The number of main seats=240

The number of balcony seats=160