Question

apply your knowledge of dilations on the coordinate plane to answer 8 - 10. 10. the coordinates of triangle xyz are given below. ronnie is going to dilate the triangle using the rule (x, y)→(1/3x, 1/3y). x(6, 3) y(9, 9) z(9, 12). a. will xyz be larger or smaller than xyz? larger / smaller because 1/3 is less than / more than 1. b. find the location of z. z(__, __). 11. max wrote the rule (x, y)→(5x, 5y) to represent the dilation below. is maxs rule correct? explain. 12. each student below wrote an algebraic rule to represent a transformation on the coordinate plane. circle the name of any student whose rule represents a dilation that is an enlargement. (circle two names.) roberto (x, y)→(6/5x, 6/5y) ella (x, y)→(0.5x, 0.5y) henry (x, y)→(x + 2, y + 2) maxine (x, y)→(1.1x, 1.1y)

Explanation:

Step1: Analyze dilation factor

The dilation rule is \((x,y)\to(\frac{1}{3}x,\frac{1}{3}y)\). Since the scale - factor \(\frac{1}{3}<1\), the dilated triangle \(X'Y'Z'\) will be smaller than \(\triangle XYZ\).

Step2: Find coordinates of \(Z'\)

Given \(Z(9,12)\) and the dilation rule \((x,y)\to(\frac{1}{3}x,\frac{1}{3}y)\). For the \(x\) - coordinate of \(Z'\), \(x'=\frac{1}{3}\times9 = 3\). For the \(y\) - coordinate of \(Z'\), \(y'=\frac{1}{3}\times12 = 4\). So \(Z'(3,4)\).

Step3: Identify enlargement rules

An enlargement has a scale - factor \(k>1\). For Roberto's rule \((x,y)\to(\frac{6}{5}x,\frac{6}{5}y)\), the scale - factor \(k = \frac{6}{5}=1.2>1\). For Ella's rule \((x,y)\to(0.5x,0.5y)\), the scale - factor \(k = 0.5<1\). For Henry's rule \((x,y)\to(x + 2,y + 2)\), it is a translation, not a dilation. For Maxine's rule \((x,y)\to(1.1x,1.1y)\), the scale - factor \(k = 1.1>1\). So the students whose rules represent an enlargement are Roberto and Maxine.

Answer:

a. Smaller, because \(\frac{1}{3}\) is less than 1.
b. \(Z'(3,4)\)
c. Roberto, Maxine