Question

a triangle has vertices ( x(1,1), y(5,1), z(3,4) ). after a dilation, its image has vertices ( x(3,3), y(15,3), z(9,12) ). what is the scale factor?
( a cdot \frac{1}{2} )
( b cdot 3 )
( c cdot \frac{1}{3} )
( d cdot 2 )
what remains unchanged in a figure after a dilation?
( a ) size of the figure
( b ). the shape and angles measures of the figure
( c ). the area of the figure
( d ). the position of the figure
if a figure is dilated by a scale factor of 0.25, what happens to the distances between points in the figure?
( a ). they increase by ( 25% ).
( b ). they remain unchanged.
( c ). they reduce to one - fourth their original lengths.
( d ). they increase to four times their original lengths.
a triangle has vertices ( a(1,2), b(3,4), c(5,1) ). after a dilation with a scale factor of 2, what are the coordinates of ( b )?
( a cdot (0,0) )
( b cdot (1.5,2) )
( c cdot (3,6) )
( d cdot (6,8) )
dilate ( \triangle def ) with vertices ( d(2,1), e(4,5), f(6,3) ) by a scale factor ( k = 2 ) with the origin as the center of dilation. what are the coordinates of ( f )?
( a cdot (12,6) )
( b cdot (6,12) )
( c cdot (12,9) )
( d cdot (3,12) )
a line segment is dilated with a scale factor ( k = 4 ). if the original length of the segment is 5, what is the length of the image?
( a cdot 10 )
( b cdot 25 )
( c cdot 15 )
( d cdot 20 )

Explanation:

Step1: Find the scale factor for the first question

For a dilation, if a point \( (x,y) \) is dilated to \( (x',y') \), the scale factor \( k=\frac{x'}{x}=\frac{y'}{y}\).
Take point \(X(1,1)\) and \(X'(3,3)\). Then \(k = \frac{3}{1}=3\).
Take point \(Y(5,1)\) and \(Y'(15,3)\). Then \(k=\frac{15}{5} = 3\) and \(\frac{3}{1}=3\).
Take point \(Z(3,4)\) and \(Z'(9,12)\). Then \(k=\frac{9}{3}=3\) and \(\frac{12}{4}=3\).

Step2: Analyze the second question

• Option a: Size of the figure changes in a dilation.
• Option b: In a dilation, the shape (similarity) and angle measures (since similar figures have equal - angle measures) remain unchanged.
• Option c: Area of a figure changes by the square of the scale factor in a dilation.
• Option d: Position of the figure (if the center of dilation is not on the figure) changes.

Step3: Analyze the third question

If the scale factor \(k = 0.25=\frac{1}{4}\), and if \(d\) is the original distance between two points and \(d'\) is the distance after dilation, then \(d'=k\times d\). So \(d'=\frac{1}{4}d\), which means the distances reduce to one - fourth their original lengths.

Step4: Find the coordinates of \(B'\) in the fourth question

For a dilation with scale factor \(k = 2\) and a point \(B(x,y)=(3,4)\), the formula for dilation with center at the origin is \(B'(kx,ky)\). So \(B'(2\times3,2\times4)=(6,8)\).

Step5: Find the coordinates of \(P'\) in the fifth question

For a dilation with scale factor \(k = 2\) and a point \(P(x,y)=(6,3)\), using the formula \(P'(kx,ky)\), we get \(P'(2\times6,2\times3)=(12,6)\).

Step6: Find the length of the image in the sixth question

The formula for the length of the image \(L'\) of a line segment with original length \(L\) and scale factor \(k\) is \(L'=k\times L\). Given \(k = 4\) and \(L = 5\), then \(L'=4\times5=20\).

Answer:

1. b. 3
2. b. The shape and angles measures of the figure
3. c. They reduce to one - fourth their original lengths
4. d. \((6,8)\)
5. a. \((12,6)\)
6. d. 20