<span>1) C. Paul and Sarah
2) D. I, III and IV
For the first problem, let's look at the three proposed transformations and see which one(s) work.
Mark. The points were reflected across the line y = x.
Point A (-6,8) would translate to (8,-6) which is not correct. So Mark's idea is wrong.
Paul, Reflect over the X-axis, then the Y-axis.
Point A(-6,8) => (-6,-8) => (6,-8) Correct
Point B(-2,2) => (-2,-2) => (2,-2) Correct
Point C(-8,5) => (-8,-5) => (8,-5) Correct
So Paul's translation works.
Sarah, Rotate 180 degrees around (0,0)
Point A(-6,8) => (6,-8) Correct
Point B(-2,2) => (2,-2) Correct
Point C(-8,5) => (8,-5) Correct
So Sarah's translation works.
So both Paul and Sarah have a translation that works.Looking at the available choices, the correct answer is
C. Paul and Sarah
The triangle OCD is given as congruent to the triangle JAL. That means that all the angles and lengths for each triangle matches the angle and length in the other triangle. Basically, both triangles are identical to each other. So let's see about the choices.
I. is line segment OC congruent to line segment JA? Yes. OC is the first segment in the triangle OCD and JA is the first segment in triangle JAL. So they're congruent by definition.
II. Is line segment OD congruent to line segment AL? No. Segment OD would be congruent to segment JL, or segment CD would be congruent to segment AL.
III. Is angle COD congruent to angle AJL? Yes. COD is the angle around point O which is the 1st point in triangle OCD and AJL is the angle around point J which is also the 1st point in the triangle JAL. So they're congruent by definition.
IV. Is the angle DCO congruent to angle JAL? C is the second point in triangle OCD and A is the second point in triangle JAL. The measure of the angle doesn't matter which way you're going. So they're congruent as well.
So I, III, and IV are all true with II being false. Looking at the available choices, you see that "D. I, III and IV" is the correct answer.</span>
