Question

Rotate △ABC
90 degrees (counterclockwise) about the origin to create △A′B′C′
. What are the coordinates of the following points?(3 points)
A′
(
,
)

B′
(
,
)

C′
(
,
)

Question 2
Now reflect △A′B′C′
across the y-axis to create △A"B"C"
. What are the new coordinates?(3 points)
A"
(
,
)

B"
(
,
)

C"
(
,
)

Question 3
True or False?

△ABC ≅ △A"B"C"
(1 point)
Responses

True
True

False
False
the coordinates for the triangle is:
A = (5,-1)
B = (5,-9)
C = (2,-7)

Answer 1

To find the coordinates of the transformed points through the described transformations, we'll first perform a 90-degree counterclockwise rotation about the origin, and then reflect the resulting points across the y-axis.

**Question 1: Rotate △ABC 90 degrees counterclockwise.**

The formula for rotating a point \((x, y)\) 90 degrees counterclockwise around the origin is:
\[
(x, y) \rightarrow (-y, x)
\]

Now, applying this to the original points:

- For point A \((5, -1)\):
\[
A' = (-(-1), 5) = (1, 5)
\]

- For point B \((5, -9)\):
\[
B' = (-(-9), 5) = (9, 5)
\]

- For point C \((2, -7)\):
\[
C' = (-(-7), 2) = (7, 2)
\]

Therefore, the coordinates after rotation are:
- A′ \( (1, 5) \)
- B′ \( (9, 5) \)
- C′ \( (7, 2) \)

**Question 2: Reflect △A′B′C′ across the y-axis.**

The formula for reflecting a point \((x, y)\) across the y-axis is:
\[
(x, y) \rightarrow (-x, y)
\]

Now applying reflection to the rotated points:

- For point A′ \( (1, 5) \):
\[
A" = (-1, 5)
\]

- For point B′ \( (9, 5) \):
\[
B" = (-9, 5)
\]

- For point C′ \( (7, 2) \):
\[
C" = (-7, 2)
\]

Therefore, the coordinates after reflection are:
- A" \( (-1, 5) \)
- B" \( (-9, 5) \)
- C" \( (-7, 2) \)

**Question 3: True or False? △ABC ≅ △A"B"C"**

To determine if triangles \(△ABC\) and \(△A"B"C"\) are congruent, we check if the corresponding sides have the same lengths.

The lengths of the sides of triangle \(△ABC\) can be found using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Calculating the lengths of \(△ABC\):
- Length AB:
\[
\sqrt{(5 - 5)^2 + (-1 + 9)^2} = \sqrt{0 + 8^2} = 8
\]
- Length BC:
\[
\sqrt{(5 - 2)^2 + (-9 + 7)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
- Length CA:
\[
\sqrt{(2 - 5)^2 + (-7 + 1)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}
\]

Now calculating the lengths of \(△A"B"C"\):
- Length A"B":
\[
\sqrt{(-1 - (-9))^2 + (5 - 5)^2} = \sqrt{8^2 + 0} = 8
\]
- Length B"C":
\[
\sqrt{(-9 - (-7))^2 + (5 - 2)^2} = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\]
- Length C"A:
\[
\sqrt{(-7 - (-1))^2 + (2 - 5)^2} = \sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
\]

Since the lengths of the corresponding sides of the two triangles are equal:
- AB = A"B' (8)
- BC = B"C' (\(\sqrt{13}\))
- CA = C"A (3\(\sqrt{5}\))

Thus, we conclude that **△ABC ≅ △A"B"C"**.

**Final response for Question 3: True**.