Mathematics, 19.10.2021 01:00 keegan5096

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

Which transformations of quadrilateral PQRS would result in the image of the quadrilateral being lo

Answers: 2

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Answer from: mohd591

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 (-3+4, -7)=(1, -7) , which is not in the first quadrant.

B) Reflection the quadrilateral across x=4 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 y=-x 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 (-10, 12) using above method. This point is not in the first quadrant.

E) We know intuitively(and can prove) that a reflection across the line x=3 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 $\boxed{E}$ is the only answer.