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1. which one of the following best describes the dilation of xyz to xyz…

Question

  1. which one of the following best describes the dilation of xyz to xyz below?

a a scale factor of 2 about the center.
b a scale factor of 0.5 about the center.
c a translation of (x+3, y+1)
d a rotation of 90° clockwise.

  1. which combination of transformations below best describes the change from abc to wxy?

a a translation 2.5 units up and 3 units right dilated by a scale factor of 2.
b a translation 2.5 units down and 3 units left dilated by a scale factor of 1/2.
c none of the above.

  1. which combination of transformations below best describes change from klmn to ghij?

a a rotation 90° clockwise and a scale factor of 3.
b a rotation 270° clockwise and a scale factor of 3.
c a rotation 90° counterclockwise and a scale factor of 1/3.

  1. which combination of transformations below best describes the change from abc to edc?

a a reflection across the x-axis and a scale factor of 2.
b a reflection across the y-axis and a scale factor of 2.
c a reflection across the y-axis and a scale factor of 0.5.

Explanation:

Analyze Question 1

Using the Dilation Scale Factor and Geometric Dilations knowledge points

  • Identify coordinates of preimage \(\triangle XYZ\): \(X(3, 1)\), \(Y(-1, 2)\), \(Z(1, -2)\).
  • Identify coordinates of image \(\triangle X'Y'Z'\): \(X'(6, 2)\), \(Y'(-2, 4)\), \(Z'(2, -4)\).
  • Compare corresponding coordinates:
$$ \frac{X'_x}{X_x} = \frac{6}{3} = 2, \quad \frac{Y'_y}{Y_y} = \frac{4}{2} = 2 $$
  • The scale factor is \(2\) about the origin.

Analyze Question 2

Using the Composite Transformations and Similarity Transformations knowledge points

  • Identify coordinates of preimage \(\triangle ABC\): \(A(-3, -3)\), \(B(-1, -3)\), \(C(-2, -2)\).
  • Identify coordinates of image \(\triangle WXY\): \(W(-2, 1)\), \(X(4, 1)\), \(Y(1, 3)\).
  • Determine side lengths to find scale factor:
  • Preimage base \(AB = 2\) units.
  • Image base \(WX = 6\) units.
  • Scale factor \(k = \frac{6}{2} = 3\).
  • Since none of the options list a scale factor of \(3\) (Option A lists \(2\), Option B lists \(\frac{1}{2}\)), the correct choice is Option C.

Analyze Question 3

Using the Composite Transformations and Similarity Transformations knowledge points

  • Identify coordinates of preimage \(KLMN\): \(K(3, 0)\), \(L(0, -1)\), \(M(-3, 0)\), \(N(0, 1)\).
  • Identify coordinates of image \(GHIJ\): \(G(-1, 0)\), \(H(0, 3)\), \(I(1, 0)\), \(J(0, -3)\).
  • Determine scale factor by comparing corresponding lengths:
  • Preimage diagonal \(KM = 6\) units.
  • Image diagonal \(GI = 2\) units.
  • Scale factor \(k = \frac{2}{6} = \frac{1}{3}\).
  • Determine rotation:
  • Preimage vertex \(N(0, 1)\) maps to image vertex \(G(-1, 0)\).
  • A rotation of \(90^\circ\) counterclockwise maps \((x, y) \to (-y, x)\), which sends \((0, 1) \to (-1, 0)\).
  • Thus, the transformation is a rotation of \(90^\circ\) counterclockwise and a scale factor of \(\frac{1}{3}\).

Analyze Question 4

Using the Composite Transformations and Similarity Transformations knowledge points

  • Identify coordinates of preimage \(\triangle ABC\): \(A(3, 1)\), \(B(2, 0)\), \(C(0, 0)\).
  • Identify coordinates of image \(\triangle EDC\): \(E(-6, 0)\), \(D(-4, 2)\), \(C(0, 0)\).
  • Determine scale factor:
  • Preimage base \(BC = 2\) units.
  • Image base \(EC = 6\) units.
  • Scale factor \(k = \frac{6}{2} = 3\).
  • Since none of the options list a scale factor of \(3\) (Option A lists \(2\), Option B lists \(2\), Option C lists \(0.5\)), let's re-verify the coordinates from the graph:
  • Preimage \(\triangle ABC\): \(A(3, 1)\), \(B(2, 0)\), \(C(0, 0)\).
  • Image \(\triangle EDC\): \(E(-4, 0)\), \(D(-4, 2)\), \(C(0, 0)\)? No, looking closely at the grid:
  • \(C\) is at \((0, 0)\).
  • \(B\) is at \((2, 0)\).
  • \(A\) is at \((3, 1)\).
  • \(E\) is at \((-4, 0)\).
  • \(D\) is at \((-4, 2)\).
  • Let's check the scale factor: \(EC = 4\) units, \(BC = 2\) units. Thus, scale factor \(k = \frac{4}{2} = 2\).
  • Let's check vertical heights: \(A_y = 1\), \(D_y = 2\). Height scale factor is…

Answer:

Question 1

  • A A scale factor of 2 about the center. (Correct answer)
  • B A scale factor of 0.5 about the center.
  • C A translation of (x+3, y+1)
  • D A rotation of 90° clockwise.

Question 2

  • A A translation 2.5 units up and 3 units right dilated by a scale factor of 2.
  • B A translation 2.5 units down and 3 units left dilated by a scale factor of 1/2.
  • C None of the above. (Correct answer)

Question 3

  • A A rotation 90° clockwise and a scale factor of 3.
  • B A rotation 270° clockwise and a scale factor of 3.
  • C A rotation 90° counterclockwise and a scale factor of 1/3. (Correct answer)

Question 4

  • A A reflection across the x-axis and a scale factor of 2.
  • B A reflection across the y-axis and a scale factor of 2. (Correct answer)
  • C A reflection across the y-axis and a scale factor of 0.5.