Question

1. triangle abc (not pictured) was transformed using a function rule. which of the following would create a similar figure?

a. ((x,y) \to (4x,y))
b. ((x,y) \to (x,4y))
c. ((x,y) \to (4x,4y))
d. ((x,y) \to left(4x, \frac{1}{4}y
ight))

Explanation:

Step1: Recall Similar Figures Transformation

For two figures to be similar, the transformation must be a dilation (scaling) with a constant scale factor for both \(x\) and \(y\) coordinates. A dilation multiplies both \(x\) and \(y\) by the same non - zero constant.

Step2: Analyze Each Option

• Option a: \((x,y)\to(4x,y)\). Here, the \(x\) - coordinate is multiplied by 4 and the \(y\) - coordinate remains the same. The scale factor is not the same for \(x\) and \(y\), so this is not a dilation for similarity.
• Option b: \((x,y)\to(x,4y)\). Here, the \(x\) - coordinate remains the same and the \(y\) - coordinate is multiplied by 4. The scale factor is not the same for \(x\) and \(y\), so this is not a dilation for similarity.
• Option c: \((x,y)\to(4x,4y)\). Here, both the \(x\) - coordinate and the \(y\) - coordinate are multiplied by 4. This is a dilation with a scale factor of 4, which will create a similar figure (since dilation preserves the shape, only changes the size).
• Option d: \((x,y)\to(4x,\frac{1}{4}y)\). Here, the \(x\) - coordinate is multiplied by 4 and the \(y\) - coordinate is multiplied by \(\frac{1}{4}\). The scale factors for \(x\) and \(y\) are different, so this is not a dilation for similarity.

Answer:

c. \((x,y)\to(4x,4y)\)