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6.9.4 Journal: Similar circles

Scenario: Prove That All Circles Are Similar
Instructions
View the video found on page 1 of this Journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
The Students' Conjectures:The two students have different methods for proving that all circles are similar.
1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)
Classmate Conjecture
John



Teresa





Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)

Yes just because some are small and some are big doesn't mean that they are different. And the formula for a circles circumference is C=2pi r and 2 and pi are alaway the same the only thing that changes is r.






Construct the Circles:
3. Draw two circles with the same center. Label the radius of the smaller circle r1 and the radius of the larger circle r2. Use the diagram you have drawn for questions 3 – 10. (2 points)

















4. In your diagram in question 3, draw an isosceles right triangle inscribed inside the smaller circle. Label this triangle ABC. (1 point)
5. What do you know about the hypotenuse of △ABC? (2 points)




6. In your diagram in question 3, extend the hypotenuse of △ABC so that it creates the hypotenuse of a right triangle inscribed in the larger circle. Add point Y to the larger circle so it is equidistant from X and Z. Then complete isosceles triangle XYZ. (1 point)
7. What do you know about the hypotenuse of △XYZ? (2 points)




8. How does △ABC compare with △XYZ? Explain your reasoning. (2 points)








9. Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)








Making a Decision
10. Who was right, Teresa or John? (1 point)




Further Exploration:
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5?

Answer:

[tex]4x^2 -4x - 48 = 0[/tex]

Step-by-step explanation:

Given

[tex]x= -3 \ or\ 4[/tex] -- roots

[tex]n= 4[/tex] --- leading coefficient

Required

Calculate the quadratic equation

[tex]x= -3 \ or\ 4[/tex]

Split to 2:

[tex]x = -3\ or\ x = 4[/tex]

Equate both to 0

[tex]x +3 = 0\ or\ x - 4= 0[/tex]

Multiply both together:

[tex](x + 3) * (x - 4) = 0 * 0[/tex]

[tex](x + 3) * (x - 4) = 0[/tex]

[tex]x^2 -4x +3x - 12 = 0[/tex]

[tex]x^2 -x - 12 = 0[/tex]

Multiply both sides by the leading coefficient, n

[tex]n * [x^2 -x - 12] = 0 * n[/tex]

[tex]4 * [x^2 -x - 12] = 0 * 4[/tex]

[tex]4 * [x^2 -x - 12] = 0[/tex]

Open brackets

[tex]4x^2 -4x - 48 = 0[/tex]

Answer:

A regular polygon is a flat shape whose sides are all equal and whose angles are all equal. The formula for finding the sum of the measure of the interior angles is (n - 2) * 180. To find the measure of one interior angle, we take that formula and divide by the number of sides n: (n - 2) * 180 / n.

Step-by-step explanation:

140

∘

Each interior angle of regular polygon of n sides is given by  

n

180(n−2)

​

 degree

∴ each interior angle =

9

180(9−2)

​

degree

=20×7

=140

Math,social studies,science, etc

Answer:

Step-by-step explanation:

Radius r = 12.5/2 = 6.25 cm

Height h = 7 cm

volume = πr²h = 3.14⋅6.25²⋅7= 858.6 cubic cm

Answer:B, C and D are all correct

Step-by-step explanation:

Because, A. 3 (5 + b)

B. 5 (b + 3)

C. 5b + 15

D. 15 + 5b

E. 3 ⋅ 5 + 3b

Answer:

hi

Step-by-step explanation:

1. The area formulas for a triangle and a parallelogram require multiplying base by height.

true

2 .A triangle and parallelogram with the same base and height will have the same area.

false

3. A rectangle and parallelogram with the same base and height will have the same area.

true

4. The area formula of a trapezoid requires finding the product of the bases.

false

5. The base of a triangle is always the shortest side.

false

Answer:

12

Step-by-step explanation:

Answer:

A. Yeah

Step-by-step explanation:

Answer:

la respuesta que buscas es bastante simple pero no lo sé

Step-by-step explanation:

Answer: thats the answer

Step-by-step explanation:

Answer:

it's simple just divide

Answer:

75°,

mUV = 105°, the other 2 are both 75°. This means mTU = 75°

Answer:

$ 602.7

Step-by-step explanation:

2009$ x 30% = 602.7

Answer:

x = 1.5

Step-by-step explanation:

8x + 8 - 3x - 12 = 14 - 7x

5x - 4 = 14 - 7x

-4 = 14 - 12x

-18 = -12x

x = 18/12 or 3/2

Answer: 452 students

25 x 18 = 450 + 2= 452

Answer:

C

Step-by-step explanation:

Correct on edge 2020

Answer:

The decay rate is of 0.05 = 5% per minute.

Step-by-step explanation:

Continuous exponential decay model:

The exponential equation of decay for an amount, after t units of time, is given by:

[tex]A(t) = A(0)e^{-rt}[/tex]

In which A(0) is the initial amount, e(approx 2.72) is the Euler value and r is the decay rate, as a decimal.

A pizza is removed from the oven at a temperature of 425 F.

This means that [tex]A(0) = 425[/tex]

After 15 minutes, the pizza has cooled to 200°F.

This means that [tex]A(15) = 200[/tex], that is, when [tex]t = 15, A(t) = 200[/tex]

We use this to find r.

[tex]A(t) = A(0)e^{-rt}[/tex]

[tex]200 = 425e^{-15r}[/tex]

[tex]e^{-15r} = \frac{200}{425}[/tex]

[tex]\ln{e^{-15r}} = \ln{\frac{200}{425}}[/tex]

[tex]-15r = \ln{\frac{200}{425}}[/tex]

[tex]r = -\frac{\ln{\frac{200}{425}}}{15}[/tex]

[tex]r = 0.05[/tex]

The decay rate is of 0.05 = 5% per minute.

Answer: the answer is A

Step-by-step explanation:

u need to look where the lines intercept and im doing this in my math class now so my teacher said wherever the lines intercept u need to take the points where they intercept at so for this one it would be (4,8) and the other is (0,-8)

Answer:

28 units sqr

Step-by-step explanation:

Area of Rectangle

A= Lw = 8(4) = 32

Area of Triangle

A= [tex]\frac{b(h)}{2}[/tex] = [tex]\frac{2(4)}{2}[/tex] = 4

32- 4 = 28 units sqr

The solution of the given equation (x + 4)(x - 3) = -10 is -2.

Thus, option C is correct.

Two algebraic expressions separated by an equal symbol in between them and with the same value of the expression are called equations.

Example = 2x +4 = 12

Here, 4 and 12 are constants and x is variable

Given equation:

Multiply the brackets,

(x + 4)(x - 3) = -10

[tex]x^2-3x+4x-12 = -10[/tex]

x^2 +x -12+10 =0

x^2 +x -2=0

x(x+2)-1(x+2)=0

(x+2)(x-1)=0

x= -2, x=1

Thus, option C is correct.

Learn more about equations here :

https://brainly.com/question/14686792

#SPJ4

There are several ways to answer this. All involve finding a way to calculate the area of shapes we're familiar with and using those areas to find the area of this unusual shape. I've included three different ways, all of which yield the same total area.

In the first case, you cut the shape into two shapes by drawing a perpendicular line from point C to segment AE. That will give you a square and a trapezoid. The area of the square is (2 m)(5 m) = 10 m², and the area of the trapezoid is (0.5)(9 m - 5 m)(4 m + 4 m - 2 m) = 12 m². So the area of the entire shape is 10 m² + 12 m² = 22 m².

In the second case, you cut the shape into two shapes by drawing a perpendicular line from point C to segment AB. That will give you a rectangle and a triangle. The area of the rectangle is (2 m)(9 m) = 18 m². The area of the triangle is (0.5)(4 m - 2 m)(9 m - 5 m) = 4 m². So the area of the entire shape is 18 m² + 4 m² = 22 m².

In the third case, you can imagine that this shape is a piece of a larger rectangle with sides 4 m and 9 m with an area of 36 m². The area of this shape would be the difference between 36 m² and the area of the imaginary trapezoid that fills in rest of the rectangle. That trapezoid would have an area of (0.5)(4 m - 2 m)(9 m + 5 m) = 14 m². So the area of the shape given would be 36 m² - 14 m² = 22 m².

In any case, the area of the shape is 22 m².

Answer:

32.2 feet

Step-by-step explanation:

Answer:

32.2

Step-by-step explanation:

Answer:

1.4/10=2/5

2.4/8=1/2

3.8/12=2/3

4.7/5=14/10

5.1/3=2/6

9514 1404 393

Answer:

  G.  84.5 cm

Step-by-step explanation:

Assuming the lateral area is 6760π cm², we can find the height from ...

  A = 2πrh

  r = A/(2πh) = (6760π cm²)/(2π·40 cm) 6760/80 cm = 84.5 cm

The radius of the can is 84.5 cm.

Answer:

5.25

alternate form

21/4,5/¼

Answer:

it would become 61

Step-by-step explanation:

the tenths place (.9) is greater or equal to five. if it follows the rule, the tens place (0) would increase by one.

Answer:

[tex]A(t) = 600(1.035)^8[/tex]

The ending balance is of $790.1.

Step-by-step explanation:

Compound interest:

The amount of money earned after t years, in compound interest, is given by:

[tex]A(t) = A(0)(1+r)^{t}[/tex]

In which A(0) is the initial amount of money and r is the interest rate, as a decimal.

$600 that earns 3.5% compound interest over 8 years

This means, respectively, that:

[tex]A(0) = 600, r = 0.035, t = 8[/tex]

So, the formula is:

[tex]A(t) = A(0)(1+r)^{t}[/tex]

[tex]A(t) = 600(1+0.035)^{8}[/tex]

[tex]A(t) = 600(1.035)^8[/tex]

Ending balance:

The ending balance is:

[tex]A(t) = 600(1.035)^8 = 790.1[/tex]

The ending balance is of $790.1.

Answer:

Step-by-step explanation:

9.1 h + 11.6 t = 9.98(60)

9.1 h + 11.6 t = 598.8 ........ (1)

h + t = 60 ........ (2)

h = 60 - t

9.1(60 - t) + 11.6 t = 598.8

546 - 9.1 t + 11.6 t = 598.8

2.5 t = 52.8

t = 52.8 ÷ 2.5 = 21.12 ≈ 21 (lb.)

h = 60 - 21.12 = 38.88 ≈ 39 (lb.)

Answer:

32

Step-by-step explanation: