Mathematics : asked on aingoldthompson

11.01.2021

Prove that the two circles shown below are similar.

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02.09.2022, solved by verified expert

Bhaskar Singh Bora

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1. Move circle with center X, 2 units to the right and 8 units down to produce circle with radius 6 and center at the origin

2. Dilate this circle with a factor of 1/2 (or compress this circle with a factor of 2) to produce a circle with radius 3

3. Now move this circle 4 units to the right and 2 units up to produce circle Y.

Since Circle X can be transformed into circle Y by translations and a dilation, therefore the two circles are similar.

Prove that the two circles shown below are similar., №15231183, 11.01.2021 12:53

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Mathematics

Prove that the two circles shown below are similar. Circle X is shown with a center at negative 2, 8 and a radius of 6. Circle Y is shown with a center of 4, 2 and a radius of 3.

Step-by-step answer

P Answered by Master

They are similar for their radii, and perimeters have the same ratio: 2

Step-by-step explanation:

The similarity of circles is proved, when we can compare the ratio, between the two radii and the two perimeter and it is the same.

2. The Circle X, has a radius of 6 and the circle Y has the radius of 3.

So let's do it:

$\frac{R_{x}}{R_{y}} =\frac{6}{3} =2$

3. Now let's check the Perimeters:

$\frac{2P_{x}}{2P_{y}}=\frac{2\pi*6}{2\pi*3}=2$

4. In addition to this, we can do this Geometrically, by inserting the Circle Y within the Circle X, and if dilate by the scale factor of 2 we'll have these two circumferences coincide.

Prove that the two circles shown below are similar. Circle X is shown with a center at negative 2, 8

Prove that the two circles shown below are similar. Circle X is shown with a center at negative 2, 8

Prove that the two circles shown below are similar. Circle X is shown with a center at negative 2, 8

Prove that the two circles shown below are similar. Circle X is shown with a center at negative 2, 8

Mathematics

Prove that the two circles shown below are similar. circle x is shown with a center at negative 2, 8 and a radius of 6. circle y is shown with a center of 4, 2 and a radius of 3.

Step-by-step answer

P Answered by Specialist

1

They are similar because the radius of circle x is a multiple of circle y. proof: translate smaller circle y to the location of circle x so that they are concentric (same center location (2, the translation would be moving the center of y left 2 units and up 6 units. now that the centers are concentric a dilation is need to increase the size of circle y to coincide with circle x. dilation by the scale factor, k [tex] r_{y} *k = r_{x} \\ \\ k = \frac{ r_{x} }{ r_{y} } [/tex]k = 6/3 k = 2 as you can see the scale factor, k is a ratio of the radius thus proving the circles are similar

Mathematics

Prove that the two circles shown below are similar. circle x is shown with a center at negative 2, 8 and a radius of 6. circle y is shown with a center of 4, 2 and a radius of 3.

Step-by-step answer

P Answered by Specialist

1

They are similar because the radius of circle x is a multiple of circle y. proof: translate smaller circle y to the location of circle x so that they are concentric (same center location (2, the translation would be moving the center of y left 2 units and up 6 units. now that the centers are concentric a dilation is need to increase the size of circle y to coincide with circle x. dilation by the scale factor, k [tex] r_{y} *k = r_{x} \\ \\ k = \frac{ r_{x} }{ r_{y} } [/tex]k = 6/3 k = 2 as you can see the scale factor, k is a ratio of the radius thus proving the circles are similar

Mathematics

Prove that the two circles shown below are similar. Circle B is shown with a center at negative 1, 5 and a radius of 4. Circle D is shown with a center of 7, 4 and a radius of 2.

Step-by-step answer

P Answered by Specialist

Proved

Step-by-step explanation:

Given

Circle B

(x,y) = (-1,5) --- center

r = 4 ---radius

Circle D

(x,y) = (7,4) --- center

r = 2 ---radius

Required

Prove the similarity

First, we shift circle B, 8 units right.

The rule is:

$(x,y) \to (x+8,y)$

So, we have:

$(-1,5) \to (-1+8,5)$

$(-1,5) \to (7,5)$

Next, we shift the resulting image of B, 1 unit down.

The rule is:

$(x,y) \to (x,y-1)$

So, we have:

$(7,5) \to (7,5-1)$

$(7,5) \to (7,4)$

At this point, both circles have the same center

(x,y) = (7,4)

Compress the image of B by a dilation of 1/2.

This only affects the radius.

i.e.

r =r_B * 1/2

r =4 * 1/2

r =2

Now, the have the same center and the same radii.

Proved

Mathematics

Prove that the two circles shown below are similar. Circle B is shown with a center at negative 1, 5 and a radius of 4. Circle D is shown with a center of 7, 4 and a radius of 2.

Step-by-step answer

P Answered by Master

(See explanation below for further details)

Step-by-step explanation:

A circle has two key parameters: Center and radius. They are similar due to two reasons:

1) Radii are multiples of each other.

$r_{B} = 2\cdot r_{D}$

2) A circle is a translated version of other circle, In this case, a translation formula for the center is:

$(x_{B}, y_{B}) = (x_{D} + 8, y_{D} - 1)$

A figure presenting both circles is included below.

Prove that the two circles shown below are similar.

Circle B is shown with a center at negative 1,

Mathematics

Prove that the two circles shown below are similar. (10 points)

Step-by-step answer

P Answered by Specialist

To prove that two shapes are similar if a transformation(such as dilation or rotation) can be applied to one shape to make it congruent to the other. i'll use dilation to prove that the circles are similar.the area of a circle is given by the equation [tex]a=\pi r^{2}[/tex], where r is the radius. the radius of circle b is 4, while the radius of circle d is 2. dilate circle d by doubling its radius, and you will get a circle congruent to circle b. thus, the circles are similar.

Mathematics

Prove that the two circles shown below are similar.

Step-by-step answer

P Answered by Specialist

The scale factor is $\frac{R_{A}}{R_{C}}=\frac{5}{2}$

We can say that both circles are similar.

Step-by-step explanation:

If we move the little circle to the center of the bigger circle, so the translate vector will be (3,5).

Now we realize that the bigger circle is just a dilation of the smaller circle, the scale factor is:

$R_{A}=5$

$R_{C}=2$

$\frac{R_{A}}{R_{C}}=\frac{5}{2}$

Therefore, we can say that both circles are similiar.

I hope it helps you!

Mathematics

(09.01 HC) Prove that the two circles shown below are similar. (10 points)

Step-by-step answer

P Answered by Master

Proved

Step-by-step explanation:

Given

Circle C

R = 4 --- radius

(x,y) = (-3,1) --- center

Circle E

r= 3

(x,y) = (4,9) --- center

Required

Prove their similarity

Translate circle C 8 units up;

The rule is:

$(x,y) \to (x,y+8)$

So, we have:

$(-3,1) \to (-3,1+8)$

$(-3,1) \to (-3,9)$

Translate circle C' 7 units right;

The rule is:

$(x,y) \to (x+7,y)$

So, we have:

$(x,y) \to (-3+7,9)$

$(x,y) \to (4,9)$

At this point, both circles now have the same center at (4,9)

The next step is to dilate circle E by 4/3

This will only affect radius and not the center of E

The rule is:

$r' = r * \frac{4}{3}$

$r' = 3 * \frac{4}{3}$

r' = 4

So, we have:

R = r' = 4 -- equal radii

$(x,y) \to (4,9)$ --- equal center

Proved

Mathematics

Prove that the two circles shown below are similar.

Step-by-step answer

P Answered by Specialist

Well we need to find the area of circles B and D

We’ll use the following formula,

$\pi r^2$

B)

The radius of the

is circle is 4 units,

so we plug 4 into $\pi r^2$ .

pi(4)^2

4*4 =16

16 * pi = 50.2654824574

The answer is 50 rounded to the nearest whole number.

D)

The radius is 2,

pi(2)^2

2*2 = 4

pi * 4 = 12.5663706144

Or 13 Rounded to the nearest whole number.

So now we do 50 ÷ 13 = 3.84615384615

Which is about 4 rounded to the nearest whole number.

Thus,

The two circles are similar just that circle D is multiplied by a scale factor of 4 to get circle B

Mathematics

Prove that the two circles shown below are similar. (10 points)

Step-by-step answer

P Answered by Specialist

To prove that two shapes are similar if a transformation(such as dilation or rotation) can be applied to one shape to make it congruent to the other. i'll use dilation to prove that the circles are similar.the area of a circle is given by the equation [tex]a=\pi r^{2}[/tex], where r is the radius. the radius of circle b is 4, while the radius of circle d is 2. dilate circle d by doubling its radius, and you will get a circle congruent to circle b. thus, the circles are similar.

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