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1. A circle has equation (x-2)^2 + (y+3)^2 + 4 and a line has equation 5y - 4x +20 = 0.
a. what is the distance from the center of the circle to the line?
b. What is the greatest distance from a point on the circle to the line?

2. The point (-3,2) is rotated 90 degrees clockwise around the origin to point B. Point B is then reflected in the line y=x to point C. What are the coordinates of C?

3. A right cylinder with a base radius of 3 units is inscribed in a sphere of radius 5 units. Find the total volume, in cubic units, of the space inside the sphere and outside the cylinder.

4. Triangle ABC and DEF are not similar, and they satisfy the following conditions: AB > DE BC > EF AC > DF Grogg claims that triangle ABC must have a greater area than triangle DEF does. Do you agree with Gross. If yes, prove his claim. If no, provide a counter example to Grogg's claim.

5. As shown in the diagram, X, Y, and Z are the midpoints of the sides of equilateral triangle ABC with center O. Describe a sequence of transformations that maps triangle ABC to triangle XYZ. (In particular you must map A to X, B to Y, and C to Z.) You must describe each transformation precisely. For example, if you use a reflection, you must specify the line you are reflecting over.

6. In triangle ABC, sin A : sin B : sin C = 2 : 3 : 4. Find cos(A+C).

7. The combined area of all the faces of a rectangular prism is 12, and the combined length of all its edges is 19. Find the length of a space diagonal of the prism.

8. In the diagram below, find AC.
(Diagram: Triangle ABC. AB is 5sqrt2. Angle A is 15 degrees. Angle B is 30 Degrees.

9. The symmetrical decahedron shown has a two-inch-by-two-inch square as its top face and its bottom fach. The isosceles trapezoid forming the other faces are all congruent. What is teh total surface area of the solid?

10. ABCDEFH is a rectangular prism with CD=5, AD=6, and AE=8. Find the volume of pyramid ABCH.

Thank you so much!

Jun 13, 2020
edited by personguy  Jun 13, 2020
edited by personguy  Jun 13, 2020

#1
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I'll try to get through as many problems as I can!

3. This is going to be a bit tricky to explain without a diagram, but just draw one yourself on a piece of paper and follow along.

Let's start off with finding the cylinder's height.

First, draw the cylinder's space diagonal. The space diagonal of the cylinder is going to be the circle's diameter, which equals 10.

Now draw the diameter of the cylinder's base. The diameter of the base, the space diagonal, and the height form a right triangle. Using the Pythagorean Theorem, we find that the height equals the square root of 102-62 , which is 8.

Now we can figure out the volume of the cylinder: 32$$\pi$$*8= 72$$\pi$$

Now we find the sphere's volume: $$\frac{4}{3}\pi$$ * 53

The space inside the sphere and outside the cylinder equals the sphere's volume - the cylinder's volume, which you should be able to calculate.

Jun 13, 2020
edited by thelizzybeth  Jun 13, 2020
edited by thelizzybeth  Jun 13, 2020
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1.

a) First, from the circle equation, we can see that the center of the circle is (2, -3). Now we use the distance from point to line formula to figure out the distance from the center of the circle to the line.

Formula: the distance between (x1, y1) and a line of the form ax + by + c = 0 is

$$\frac{|ax_{1}+by_{1}+c|}{\sqrt(a^2+b^2)}$$

Now, plugging the values in, $$\frac{|-8-15+20|}{\sqrt(16+25)}$$, and solving for this equals $$\frac{3}{\sqrt(41)}$$. Rationalizing the denomiator, we get that the answer is $$\frac{3\sqrt(41)}{41}$$.

b) The greatest distance is the distance we just found above plus the circle's radius. (To see why, draw a diagram- it's just a line from the end of the circle to the point)

However, your equation implies that the circle's radius is the square root of -4, which is not a real number. Did you type it wrong?

Jun 13, 2020
#3
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2. We start off by rotating the point (-3, 2) 90 degrees clockwise around the origin. Now the coordinates are (2,3). Reflecting this point over the line y=x means flipping the x and y values. Therefore, the coordinates of this new point C is (3,2).

I did not explain the WHY of the process, but Khan Academy has an entire unit about translations, reflections, and rotations. Check out this link: https://www.khanacademy.org/math/basic-geo/basic-geo-transformations-congruence

Jun 13, 2020