Question

which statement is true?
a. adam took at least 25 seconds to swim one lap.
b. adam took at most 21 seconds to swim one lap.
c. half of the time, adam took less than 31 seconds to swim one lap.
d. one - fourth of the time, adam took less than 25 seconds to swim one lap.
e. one - fourth of the time, adam took more than 34 seconds to swim one lap.

2
the coordinates of the end points of a linear segment ab are a(6,10) and b(8,4). the line segment ab is translated 3 units to the left and 2 units up and then reflected over the line y = -x to obtain line segment ab.
what are the coordinates of the end points of line segment ab?
a. a(12,3) and b(6,5)
b. a(3,12) and b(5,6)
c. a(3, - 12) and b(5, - 6)
d. a(-3,12) and b(-5,6)
e. a(-12,-3) and b(-6,-5)

Explanation:

Step1: Translate the endpoints

For point $A(6,10)$, translating 3 units to the left and 2 units up:
The $x$-coordinate of $A$ becomes $6 - 3=3$, and the $y$-coordinate becomes $10 + 2 = 12$, so the new - point is $A_1(3,12)$.
For point $B(8,4)$, translating 3 units to the left and 2 units up:
The $x$-coordinate of $B$ becomes $8 - 3 = 5$, and the $y$-coordinate becomes $4+2 = 6$, so the new - point is $B_1(5,6)$.

Step2: Reflect the translated points over the line $y=-x$

The rule for reflecting a point $(x,y)$ over the line $y =-x$ is $(x,y)\to(-y,-x)$.
For point $A_1(3,12)$, after reflection over $y=-x$, the $x$-coordinate becomes $-12$ and the $y$-coordinate becomes $-3$, so $A'(-12,-3)$.
For point $B_1(5,6)$, after reflection over $y=-x$, the $x$-coordinate becomes $-6$ and the $y$-coordinate becomes $-5$, so $B'(-6,-5)$.

Answer:

E. $A'(-12,-3)$ and $B'(-6,-5)$