Question

Use the image to answer the question.

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Three triangles are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis 2 comma negative 2 right parenthesis, upper B left parenthesis 2 comma negative 4 right parenthesis, and upper C left parenthesis 5 comma negative 4 right parenthesis. The second triangle, placed in the second quadrant, has its vertices marked with closed points at left parenthesis negative 3 comma 6 right parenthesis, left parenthesis negative 5 comma 6 right parenthesis, and left parenthesis negative 5 comma 3 right parenthesis. The third triangle, placed in the third quadrant, has its vertices marked with closed points at left parenthesis negative 2 comma negative 2 right parenthesis, left parenthesis negative 4 comma negative 2 right parenthesis, and left parenthesis negative 4 comma negative 5 right parenthesis. An arrow goes from the point left parenthesis negative 2 comma negative 2 right parenthesis to left parenthesis negative 3 comma 6 right parenthesis; an arrow goes from the point left parenthesis negative 4 comma negative 2 right parenthesis to left parenthesis negative 5 comma 6 right parenthesis; an arrow goes from left parenthesis negative 4 comma negative 5 right parenthesis to left parenthesis negative 5 comma 3 right parenthesis.

What sequence of transformations is depicted in the figure, proving the congruence of the triangles?

(2 points)
rotation of (
) degrees clockwise about the origin, then translation of (
,
​​​​​​​)​​​​​​​
Remaining Attempts : 3
Skip to navigation

Answers

Answered by GPT-4o mini
To determine the transformations that prove the congruence of the triangles depicted in the figure, we analyze the coordinates of the vertices of each triangle.

Let's take the two triangles in the second and third quadrants.

1. **Identify triangle in the third quadrant**:
- Triangle vertices:
- \( A(-2, -2) \)
- \( B(-4, -2) \)
- \( C(-4, -5) \)

2. **Identify triangle in the second quadrant**:
- Triangle vertices:
- \( D(-3, 6) \)
- \( E(-5, 6) \)
- \( F(-5, 3) \)

To find how triangle ABC is transformed to triangle DEF, we analyze the movement:

- The vertices are moved from lower Y-values (negative in the third quadrant) to higher Y-values (positive in the second quadrant).

**1. Rotation:**
- The triangles are situated in different quadrants, indicating a possible rotation.

The movement from:
- \( A(-2, -2) \) to \( D(-3, 6) \) suggests a rotation of 180 degrees clockwise about the origin, which would effectively switch the X and Y coordinates while changing their signs.

**2. Translation:**
- After the rotation, we observe a shift:
- From \( D(-3, 6) \) to \( E(-5, 6) \) and \( E(-5, 6) \) to \( F(-5, 3) \).

Looking at a translation, it appears that every vertex shifts accordingly after the rotation has been completed.

So to summarize the transformations:

- **Rotation**: **180 degrees clockwise about the origin**
- **Translation**: Move down and left to match the congruent corresponding points, specifically translating by \( (-1, 0) \) which signifies a 1 unit left (change in x-coordinates) and no change in y-coordinate.

Therefore, the complete answer is:
- Rotation of **180 degrees clockwise** about the origin, then translation of **(-1, 0)**.
There are no human answers yet. A form for humans to post answers is coming very soon!

Related Questions