Question

spaced practice

1. give the coordinates of △abc after a transformation of △abc with the coordinates a(6, - 3), b(9, 5), and c(5, 6). use the origin as the center of dilation or rotation, as needed.

a. dilate △abc by a scale factor of $\frac{1}{3}$.
b. dilate △abc by a scale factor of 4.
c. rotate △abc 180 degrees.
d. reflect △abc across the x - axis.
in the equation y = 4x + 7, x is the number of items and y is the total cost. what is the unit rate? include units in your response

Explanation:

Step1: Recall dilation formula

For a dilation with scale - factor \(k\) centered at the origin, the transformation of a point \((x,y)\) is \((kx,ky)\).

a. Dilate \(\triangle ABC\) by a scale factor of \(\frac{1}{3}\)

• For point \(A(6,-3)\): \(A'=(6\times\frac{1}{3},-3\times\frac{1}{3})=(2, - 1)\)
• For point \(B(9,5)\): \(B'=(9\times\frac{1}{3},5\times\frac{1}{3})=(3,\frac{5}{3})\)
• For point \(C(5,6)\): \(C'=(5\times\frac{1}{3},6\times\frac{1}{3})=(\frac{5}{3},2)\)

b. Dilate \(\triangle ABC\) by a scale factor of \(4\)

• For point \(A(6,-3)\): \(A'=(6\times4,-3\times4)=(24,-12)\)
• For point \(B(9,5)\): \(B'=(9\times4,5\times4)=(36,20)\)
• For point \(C(5,6)\): \(C'=(5\times4,6\times4)=(20,24)\)

Step2: Recall rotation formula

For a \(180^{\circ}\) rotation about the origin, the transformation of a point \((x,y)\) is \((-x,-y)\).

c. Rotate \(\triangle ABC\) \(180\) degrees

• For point \(A(6,-3)\): \(A'=(-6,3)\)
• For point \(B(9,5)\): \(B'=(-9,-5)\)
• For point \(C(5,6)\): \(C'=(-5,-6)\)

Step3: Recall reflection formula

For a reflection across the \(x\) - axis, the transformation of a point \((x,y)\) is \((x,-y)\).

d. Reflect \(\triangle ABC\) across the \(x\) - axis

• For point \(A(6,-3)\): \(A'=(6,3)\)
• For point \(B(9,5)\): \(B'=(9,-5)\)
• For point \(C(5,6)\): \(C'=(5,-6)\)

Answer:

a. \(A'(2,-1)\), \(B'(3,\frac{5}{3})\), \(C'(\frac{5}{3},2)\)
b. \(A'(24,-12)\), \(B'(36,20)\), \(C'(20,24)\)
c. \(A'(-6,3)\), \(B'(-9,-5)\), \(C'(-5,-6)\)
d. \(A'(6,3)\), \(B'(9,-5)\), \(C'(5,-6)\)