Triangle ABC is translated 2 units up and 4 units to the right and is then rotated 180° to produce the image triangle A′B′C′. What are the coordinates of B′ in the resulting figure?
A. (−8,0)
B. (−1,−6)
C. (−6,1)
D. (−5,−10)
Let the original coordinates of point B be $(x,y)$. After the translation, its new coordinates will be $(x+4,y+2)$. Finally, after a 180 degree rotation, its new coordinates will be $(-x-4,-y-2)$. Therefore, the coordinates of $B'$ are $(-x-4,-y-2)$.
We don't know the original coordinates of point B, so let's look at the answer choices and work backwards. We can plug in each answer choice as the coordinates of $B'$ and work backwards to see if it satisfies the given transformations.
(A) If $B' = (-8,0)$, then the coordinates of $B$ before rotation would be $(4,-2)$, and the coordinates of $B$ before translation would be $(0,-4)$. But these coordinates don't fit a translation upward by 2 units and to the right by 4 units, so this answer choice doesn't work.
(B) If $B' = (-1,-6)$, then the coordinates of $B$ before rotation would be $(5,4)$, and the coordinates of $B$ before translation would be $(1,2)$. This fits the given transformations, so this is a possible answer.
(C) If $B' = (-6,1)$, then the coordinates of $B$ before rotation would be $(2,-4)$, and the coordinates of $B$ before translation would be $(-2,-6)$. But these coordinates don't fit a translation upward by 2 units and to the right by 4 units, so this answer choice doesn't work.
(D) If $B' = (-5,-10)$, then the coordinates of $B$ before rotation would be $(9,12)$, and the coordinates of $B$ before translation would be $(5,10)$. But these coordinates don't fit a translation upward by 2 units and to the right by 4 units, so this answer choice doesn't work.
Therefore, the answer is $\boxed{\textbf{(B)}\ (-1,-6)}$.
Give me a quote from the Phantom tollbooth to help support your thinking (Cite where in the novel this quote came from…chapter, page)
"Expect everything, I always say, and the unexpected never happens." - This quote is from Chapter 3 of The Phantom Tollbooth on page 49. It encourages open-mindedness and creativity, and supports the idea that being prepared for anything can help prevent unexpected surprises.
Explain how the quote helps support your thinking
The quote "Expect everything, I always say, and the unexpected never happens" suggests that by being open-minded and ready for any possibility, one can avoid being surprised by unexpected events. This quote supports the idea that it's important to approach problems and challenges with a flexible and creative mindset. In math, this means being prepared to approach problems from different angles and to try different methods until you find a solution. The quote encourages learners to embrace the unknown and to be prepared for whatever might come their way, helping them to think critically and creatively in order to find the answers they need.
To find the coordinates of B' in the resulting figure, we need to follow the given transformations step by step.
1. Translation: The triangle is translated 2 units up and 4 units to the right. To find the new coordinates after the translation, add 2 to the y-coordinate and add 4 to the x-coordinate of each point.
Let's assume the coordinates of B in the original triangle ABC are (x, y). After the translation, the new coordinates of B will be (x + 4, y + 2).
2. Rotation: The triangle is rotated 180°. To rotate a point (x, y) by 180°, we negate both the x and y coordinates.
The new coordinates of B' after rotation can be found by multiplying the translated coordinates (x + 4, y + 2) by -1.
B' = (-(x + 4), -(y + 2))
= (-x - 4, -y - 2)
Now, we need to compare the resulting coordinates of B' to the option choices and find the matching coordinates.
Options:
A. (−8,0)
B. (−1,−6)
C. (−6,1)
D. (−5,−10)
Comparing the calculated coordinates (-x - 4, -y - 2) with the given options, we can see that option A (-8, 0) matches.
Therefore, the coordinates of B' in the resulting figure are (-8, 0). Hence, the answer is A. (−8,0).