Question

According to these three facts, which statements are true?

Answer 1

Q1) We have the following statements:

1. Circle W has center (−3, 0) and radius 8.

We can write this statement as the following equation:

$(x-h)^2+(y-k)^2=r^2 \therefore (h,k)=(-3,0) \ and \ r=8 \\ \\ \therefore (x+3)^2+y^2=64$

The graph of this equation is shown in Figure 1.

2. Circle V is a translation of circle W, 2 units down.

To do this translation we add two units to the y-coordinate, so:

$(x+3)^2+(y+2)^2=64$

The graph is shown in Figure 2.

3. Circle V is a dilation of circle W with a scale factor of 2.

To do this dilatation we multiply both the center and the radius by the scale factor of 2, thus:

$(x+3)^2+y^2=64 \\ (h,k)=(-3,0) \\ (h_v,k_v)=2\times (-3,0)=(-6,0) \ and \ r_v=8\times2=16 \\ \\ (x+6)^2+y^2=256$

This circle is shown in Figure 3

So let's analyze each statement.

Q1.1) The center of circle V is (−5, 0).

This is false. From the statement 2 we know that the new center is (-3, -2) and from 3 the new center is (-6, 0).

Q1.2) Circle V and circle W are similar.

Since all circles have the same shape even though they may be different sizes, then all circles are similar. Therefore, it is true that circle V and circle W are similar.

Q1.3) The radius of circle V is 16.

From the statement 3 we can affirm that this is true. By applying the dilatation with a scale factor of 2 we find out that the radius of the new circle is in fact equal to 16.

Q1.4) Circle V and circle W have the same center. 

From the statement 2 we know that the new center is (-3, -2) and from 3 the new center is (-6, 0). On the other hand, the center of circle W is (-3, 0). From this, it follows that this statement is false.

Q2) Suppose x is any positive number.

We have two concentric circles as follows:

$x\ \textgreater \ 0 \\ \\ Circle \ 1: Center \ (0,0) \ and \ radius \ 2x \\ \\ Circle \ 2: Center \ (0, 0) \ and \ radius \ 10x$

Q2.1) Why is circle 1 similar to circle 2?

If we perform a dilatation, that is, a resizing of one circle, centered on the shared center, until both circles overlap, it will be true that the circles will always overlap no matter what size they are, thus, they are similar. In fact, as we said in above all circles are similar. 

Q2.2) Circle 2 is a dilation of circle 1 with a scale factor of 0.2.

Suppose that:

$x=2$

Then circle 1 is given by the following equation:

$x^2+y^2=4$

If we dilate this circle with a scale factor of 0.2 then:

$Circle \ 1: x^2+y^2=4 \\ Diameter \ 1=2r=2\times 2=4 \\ \\ (h,k)=(0,0) \\ (h_v,k_v)=0.2\times (0,0)=(0,0) \ and \ r_v=0.2\times 2=0.4 \\ \\ Circle \ 2:x^2+y^2=0.16 \\ Diameter \ 2=2\times 0.4=0.8 \\ \\ So: \\ \\ Diameter \ 1 \neq Diameter \2$

So the statement both circles have congruent diameters is false.

Q2.3) Circle 2 is a dilation of circle 1 with a scale factor of 5.

We have the same equation for circle 1:

$x^2+y^2=4$

$Circle \ 1: x^2+y^2=4 \\ \\ Dilatation: \\ (h,k)=(0,0) \\ (h_v,k_v)=5\times (0,0)=(0,0) \ and \ r_v=5\times 2=10 \\ \\ Circle \ 2:x^2+y^2=100$

Two circles have the same area if they have the same radius. Given that this is no applied to our circles, the statement circle 1 and circle 2 have equal areas is false.