Answered

Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

After a rotation of 90° about the origin, the coordinates of the vertices of the image of a triangle are A(6, 3), B(-2, 1), and C(1, 7). What are the coordinates of the vertices of the pre-image?

Sagot :

GinnyAnswer
To find the coordinates of the vertices of the pre-image of a triangle after a 90° counterclockwise rotation about the origin, we need to reverse the transformation. A 90° counterclockwise rotation involves the transformation of a point [tex]\((x, y)\)[/tex] to [tex]\((-y, x)\)[/tex]. To find the pre-image, we use the inverse transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex].

Given the coordinates of the vertices of the image after rotation:
- [tex]\(A' (6, 3)\)[/tex]
- [tex]\(B' (-2, 1)\)[/tex]
- [tex]\(C' (1, 7)\)[/tex]

We will now apply the inverse transformation [tex]\((x, y) \rightarrow (y, -x)\)[/tex] to each of these vertices.

### Step 1: Finding the pre-image of [tex]\(A'\)[/tex]
For [tex]\(A' (6, 3)\)[/tex]:
- [tex]\(x = 6\)[/tex]
- [tex]\(y = 3\)[/tex]

Applying the inverse transformation:
[tex]\[ (6, 3) \rightarrow (3, -6) \][/tex]

So, the coordinates of [tex]\(A\)[/tex] are [tex]\((3, -6)\)[/tex].

### Step 2: Finding the pre-image of [tex]\(B'\)[/tex]
For [tex]\(B' (-2, 1)\)[/tex]:
- [tex]\(x = -2\)[/tex]
- [tex]\(y = 1\)[/tex]

Applying the inverse transformation:
[tex]\[ (-2, 1) \rightarrow (1, 2) \][/tex]

So, the coordinates of [tex]\(B\)[/tex] are [tex]\((1, 2)\)[/tex].

### Step 3: Finding the pre-image of [tex]\(C'\)[/tex]
For [tex]\(C' (1, 7)\)[/tex]:
- [tex]\(x = 1\)[/tex]
- [tex]\(y = 7\)[/tex]

Applying the inverse transformation:
[tex]\[ (1, 7) \rightarrow (7, -1) \][/tex]

So, the coordinates of [tex]\(C\)[/tex] are [tex]\((7, -1)\)[/tex].

### Conclusion
The coordinates of the vertices of the pre-image of the triangle are:
- [tex]\(A (3, -6)\)[/tex]
- [tex]\(B (1, 2)\)[/tex]
- [tex]\(C (7, -1)\)[/tex]

Thus, after reversing the 90° counterclockwise rotation, the original coordinates of the vertices of the pre-image are:
[tex]\[ \boxed{(3, -6), (1, 2), (7, -1)} \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.