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3 years ago

9.01 prove that the two circles are similar

Mathematics

1 answer:

il63 [147K]3 years ago

4 0

Answer:

see the procedure

Step-by-step explanation:

we know that

Figures can be proven similar if one, or more, similarity transformations (reflections, translations, rotations, dilations) can be found that map one figure onto another.

In this problem to prove circle C and circle A are similar, a translation and a scale factor (from a dilation) will be found to map one circle onto another.

we have that

Circle C is centered at (0,-1) and has a radius of 2 units

Circle A is centered at (3,4) and has a radius of 5 units

step 1

Move the center of the circle C onto the center of the circle A

the transformation has the following rule

(x,y)--------> (x+3,y+5)

(0,-1)------> (0+3,-1+5)-----> (3,4)

center circle C is now equal to center circle A

The circles are now concentric (they have the same center)

step 2

A dilation is needed to increase the size of circle C to coincide with circle A

scale factor=radius circle A/radius circle C-----> 5/2=2.5

radius circle C will be=2*scale factor-----> 2*2.5=5 units

radius circle C is now equal to radius circle A

therefore

A translation, followed by a dilation will map one circle onto the other, thus proving that the circles are similar

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Question #1

Part A:
The y-intercept can be found when x = 0. If you look at your table, when x = 0, y = 5. So the y-intercept is 5.

Part B:
$\sf Slope = \frac{27-5}{1-0} = \frac{22}{1} = \boxed{22}$
The slope is 22.

Part C:
y = mx + b
y = 22x + 5

We are given 225 as the range, or in place of y.
225 = 22x + 5
220 = 22x
x = 10

The domain is 10.

Question #2

Part A:

(2,255)
(5,480)

Standard form is Ax + By = C

$\sf Slope = \frac{y_2-y_1}{x_2-x_1} = \frac{480-225}{5-2} = \frac{255}{3} = \boxed{85}$

Let's plug this into this form first:
$y - y_1 = m(x-x_1)\\\\y -225 = 85(x-2)\\\\y - 225 = 85x - 170\\\\y = 85x + 55$

Now, let's make it into Standard Form.
$y = 85x + 55\\\\y - 85x + 55\\\\ -85x + y = 55\\\\ -1(-85x +y) = -1(55)\\\\\boxed{85x - y = -55}$
What, which is in the box, is your final answer. :)

Part B:
Function notation simply means replacing y with f(x).
We had y = 85x + 55
So your answer is:
$\boxed{f(x) = 85x + 55}$

Part C:
Using the final answer which we got in Part A, we would know that the y-intercept is (0,55) and the x-intercept is (-55/85, 0). We would plot these 2 points, and then draw a line between them. :)

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