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 Skip to navigation

the coordinates for the triangle are
a = 5, -1
b = 5, -9
c = 2, -7

Answer 1

To solve the problem step by step, let's start with the coordinates of triangle \( ABC \):

- A(5, -1)
- B(5, -9)
- C(2, -7)

**Question 1: Rotate \( \triangle ABC \) 90 degrees counterclockwise about the origin to create \( \triangle A'B'C' \).**

For a 90-degree counterclockwise rotation about the origin, the transformation of a point \((x, y)\) to \((x', y')\) is given by:
\[
(x', y') = (-y, x)
\]

Applying this transformation to each point:

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

2. **For B(5, -9)**:
\[
B' = (-(-9), 5) = (9, 5)
\]

3. **For C(2, -7)**:
\[
C' = (-(-7), 2) = (7, 2)
\]

So the coordinates for \( \triangle A'B'C' \) are:
- A' (1, 5)
- B' (9, 5)
- C' (7, 2)

**Question 2: Reflect \( \triangle A'B'C' \) across the y-axis to create \( \triangle A"B"C" \).**

For reflection across the y-axis, the transformation is given by:
\[
(x', y') = (-x, y)
\]

Applying this transformation to each point:

1. **For A' (1, 5)**:
\[
A" = (-1, 5)
\]

2. **For B' (9, 5)**:
\[
B" = (-9, 5)
\]

3. **For C' (7, 2)**:
\[
C" = (-7, 2)
\]

So the coordinates for \( \triangle A"B"C" \) are:
- A" (-1, 5)
- B" (-9, 5)
- C" (-7, 2)

**Question 3: True or False? \( \triangle ABC \cong \triangle A"B"C" \)**

To determine if the triangles are congruent, we need to check if all corresponding sides are equal in length.

The original triangle \( ABC \):
- A(5, -1) to B(5, -9):
\[
|y_2 - y_1| = |-9 - (-1)| = 8
\]
- B(5, -9) to C(2, -7):
\[
\sqrt{(2-5)^2 + (-7 + 9)^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
- C(2, -7) to A(5, -1):
\[
\sqrt{(5 - 2)^2 + (-1 + 7)^2} = \sqrt{(3)^2 + (6)^2} = \sqrt{9 + 36} = \sqrt{45}
\]

The lengths of the sides of \( \triangle ABC \) are \( 8, \sqrt{13}, \sqrt{45} \).

Now for \( \triangle A"B"C" \):
- A"(-1, 5) to B"(-9, 5):
\[
|x_2 - x_1| = |-9 - (-1)| = 8
\]
- B"(-9, 5) to C"(-7, 2):
\[
\sqrt{(-7 + 9)^2 + (2 - 5)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]
- C"(-7, 2) to A"(-1, 5):
\[
\sqrt{(-1 + 7)^2 + (5 - 2)^2} = \sqrt{(6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45}
\]

The lengths of the sides of \( \triangle A"B"C" \) are also \( 8, \sqrt{13}, \sqrt{45} \).

Since all corresponding sides are equal, the triangles are congruent.

**Final responses:**
1. A' (1, 5), B' (9, 5), C' (7, 2)
2. A" (-1, 5), B" (-9, 5), C" (-7, 2)
3. True