Question

1. if a figure is dilated by a scale factor of 0.25, what happens to the distances between points in the figure?

a. they reduce to one - fourth their original lengths.
b. they increase to four times their original lengths.
c. they increase by 25%.
d. they remain unchanged.

1. which term refers to the multiplier used to realize a figure in a dilation?

a. ratio
b. proportion
c. preimage
d. scale factor

1. a line segment with endpoints ( p(1, 2) ) and ( q(3, 4) ) is dilated with a scale factor of ( \frac{1}{2} ). what are the coordinates of ( q )?

a. ( (1.5, 2) )
b. ( (1.5, 3) )
c. ( (0, 8) )
d. ( (2, 3) )

Explanation:

Question 4

Dilation is a transformation that changes the size of a figure. The scale factor determines how the distances (and lengths) of the figure change. A scale factor of 0.25 (which is $\frac{1}{4}$) means that each distance is multiplied by 0.25, so distances reduce to one - fourth of their original lengths. Option b is incorrect because a scale factor less than 1 should decrease lengths, not increase. Option c is incorrect as a 25% increase would correspond to a scale factor of 1.25, not 0.25. Option d is incorrect because dilation changes the size, so distances do not remain unchanged.

In a dilation, the multiplier that is used to resize the figure is called the scale factor. A ratio is a comparison of two quantities, a proportion is an equation stating that two ratios are equal, and the pre - image is the original figure before dilation. So the term for the multiplier in dilation is the scale factor.

Step 1: Recall the dilation formula for a point \((x,y)\) with scale factor \(k\) centered at the origin (assuming the dilation is centered at the origin, which is a common case if not specified otherwise). The formula for the image of the point \((x,y)\) after dilation is \((kx,ky)\).

The original coordinates of \(Q\) are \((3,4)\) and the scale factor \(k=\frac{1}{2}\).

Step 2: Apply the dilation formula to the \(x\) - coordinate of \(Q\).

For the \(x\) - coordinate: \(x'=k\times x=\frac{1}{2}\times3 = 1.5\)

Step 3: Apply the dilation formula to the \(y\) - coordinate of \(Q\).

For the \(y\) - coordinate: \(y'=k\times y=\frac{1}{2}\times4 = 2\)? Wait, no, wait. Wait, maybe the dilation is centered at point \(P\)? Wait, if we assume the dilation is centered at the origin, the calculation above is for origin - centered dilation. But let's re - check. Wait, the line segment has endpoints \(P(1,2)\) and \(Q(3,4)\). If we consider dilation centered at the origin, the image of \(Q(3,4)\) with scale factor \(\frac{1}{2}\) is \((\frac{1}{2}\times3,\frac{1}{2}\times4)=(1.5,2)\)? But that's option a. Wait, but maybe the dilation is centered at \(P\). Let's check that case. The vector from \(P\) to \(Q\) is \((3 - 1,4 - 2)=(2,2)\). If we dilate with scale factor \(\frac{1}{2}\) centered at \(P\), the new vector from \(P\) to \(Q'\) is \(\frac{1}{2}\times(2,2)=(1,1)\). Then the coordinates of \(Q'\) are \(P+(1,1)=(1 + 1,2+1)=(2,3)\)? No, that's option d. Wait, there is a confusion here. But in most basic dilation problems, if the center is not specified, it is assumed to be the origin. Wait, but let's re - calculate with origin - centered dilation: \(Q(3,4)\) with scale factor \(\frac{1}{2}\): \(x = 3\times\frac{1}{2}=1.5\), \(y = 4\times\frac{1}{2}=2\), so \((1.5,2)\) which is option a. But wait, maybe the problem has a typo or I misread. Wait, the options: a is \((1.5,2)\), b is \((1.5,3)\), c is \((0,8)\), d is \((2,3)\). Wait, maybe the dilation is centered at \(P\). Let's recast: The center of dilation is \(P(1,2)\). The formula for dilation centered at \((a,b)\) with scale factor \(k\) is \((a + k(x - a),b + k(y - b))\). So for \(Q(3,4)\), \(a = 1\), \(b = 2\), \(k=\frac{1}{2}\). Then \(x'=1+\frac{1}{2}(3 - 1)=1+\frac{1}{2}\times2=1 + 1 = 2\), \(y'=2+\frac{1}{2}(4 - 2)=2+\frac{1}{2}\times2=2 + 1 = 3\). So \(Q'\) is \((2,3)\), which is option d. But this is a bit confusing. Wait, maybe the original problem assumes origin - centered dilation. But let's check the scale factor \(\frac{1}{2}\). If we take origin - centered, \(Q(3,4)\) becomes \((1.5,2)\) (option a). But maybe the problem has a different center. Wait, the problem says "a line segment with endpoints \(P(1,2)\) and \(Q(3,4)\) is dilated with a scale factor of \(\frac{1}{2}\)". In many cases, when a line segment is dilated and no center is specified, it can be assumed that the center is the origin, but sometimes it can be the mid - point or one of the endpoints. Wait, if we take the center as \(P\), then as we calculated, \(Q'\) is \((2,3)\) (option d). But let's check the calculation again. The vector from \(P\) to \(Q\) is \((3 - 1,4 - 2)=(2,2)\). Dilation with scale factor \(\frac{1}{2}\) would make the vector \((1,1)\), so adding to \(P(1,2)\) gives \((2,3)\). So the answer is d.

Answer:

a. They reduce to one - fourth their original lengths.

Question 5