Question

solve the problems below. be sure to show your thinking.

1. which of the following algebraic rules represents a dilation that is an enlargement?

a. $(\frac{1}{3}x,\frac{1}{3}y)$
b. $(0.1x,0.1y)$
c. $(\frac{5}{6}x,\frac{5}{6}y)$
d. $(\frac{5}{2}x,\frac{5}{2}y)$

1. a triangle in quadrant ii is going to be reflected over the y - axis. which of the following is a true statement?

a. the image will be in quadrant iii.
b. the orientation of the figure will not change.
c. the orientation of the vertices will change.
d. the image and the pre - image will not be congruent.

Explanation:

Question 13

Step1: Recall dilation rule

A dilation is an enlargement if the scale factor \(k> 1\). For a rule \((kx,ky)\), check the value of \(k\).

Step2: Analyze each option

• Option A: \(k = \frac{1}{3}\approx0.33<1\) (reduction)
• Option B: \(k = 0.1<1\) (reduction)
• Option C: \(k=\frac{5}{6}\approx0.83<1\) (reduction)
• Option D: \(k=\frac{5}{2} = 2.5>1\) (enlargement)

Step1: Reflect over y - axis rule

The rule for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\). A triangle in quadrant II \((x<0,y > 0)\) will go to quadrant I \((x>0,y>0)\) after reflection over the \(y\) - axis.

Step2: Analyze orientation and congruence

• Orientation: Reflection changes the orientation of the vertices (e.g., clockwise to counter - clockwise or vice - versa)
• Congruence: Reflection is a rigid transformation, so the image and pre - image are congruent

Step3: Analyze each option

• Option A: Image is in quadrant I (not III), so A is wrong
• Option B: Orientation of vertices changes (reflection is a non - direct isometry), so B is wrong
• Option C: Orientation of vertices changes (correct)
• Option D: Image and pre - image are congruent (reflection is a rigid motion), so D is wrong

Answer:

D. \((\frac{5}{2}x,\frac{5}{2}y)\)

Question 14