Question

dilations
similarity activity
answer the following questions. make sure to explain all your work and show all your steps.

1. analyze the following triangle.

2a: identify the vertices of this triangle.
2b: the triangle is being dilated by a factor of 1/2.
what are the new vertices of this triangle? show all your work.

Explanation:

Step1: Identify vertices

To find the vertices of a point on a coordinate - plane, we use the \(x\) (horizontal) and \(y\) (vertical) coordinates.
For point \(A\): The \(x\) - coordinate is \(3\) and the \(y\) - coordinate is \(2\), so \(A=(3,2)\).
For point \(B\): The \(x\) - coordinate is \(6\) and the \(y\) - coordinate is \(3.5\), so \(B=(6,3.5)\).
For point \(C\): The \(x\) - coordinate is \(8\) and the \(y\) - coordinate is \(0\), so \(C=(8,0)\).

Step2: Apply dilation formula

The formula for dilation about the origin \((x,y)\to(kx,ky)\), where \(k = \frac{1}{2}\).
For point \(A=(3,2)\):
\(x'=\frac{1}{2}\times3=\frac{3}{2}=1.5\), \(y'=\frac{1}{2}\times2 = 1\), so \(A'=(1.5,1)\).
For point \(B=(6,3.5)\):
\(x'=\frac{1}{2}\times6 = 3\), \(y'=\frac{1}{2}\times3.5=1.75\), so \(B'=(3,1.75)\).
For point \(C=(8,0)\):
\(x'=\frac{1}{2}\times8 = 4\), \(y'=\frac{1}{2}\times0=0\), so \(C'=(4,0)\).

Answer:

2A: The vertices of the triangle are \(A=(3,2)\), \(B=(6,3.5)\), and \(C=(8,0)\).
2B: The new vertices after dilation with a scale factor of \(\frac{1}{2}\) are \(A'=(1.5,1)\), \(B'=(3,1.75)\), and \(C'=(4,0)\).