Honors Geometry Problems

[Twins]

Please can you help me please.
P.S. I don't know how to send the image of the graph. but the circle lies on at 5's
5 and -5 on x axis and 5 and -5 on y axis. And point 0,0 the center of the circle.
Also the graph for #1 is the same kind of graph for #2

1. Consider a circle centered at the origin. Point A on the circle has the coordinates (3,4).

a) What is the slope if the radius drawn to the point A?
b) What is the slope of the tangent line through point A?
c) Write an equation for the line tangent to the circle at point A?


2. Circle O centered at the origin passes through points A(3,4) and point B(4,-3). Write an equation for the perpendicular bisector of segment AB. Sketch the line on the graph.



3. The diameter if Earth is approximately 8000 miles.

a) How many miles would be in a trip around Earth's equator?
b) Astronaunt Polly Hedra circles Earth every 90 minutes in a path above the equator. What distance along the equator will she pass directly over while eating a quick 15 minute lunch?
c) The radius of the moon is approximately 1087 miles. If Polly Hedra travels at the same speed, how long would it take her to make one complete trip around the moon's equator?

4. A hang glider is flying at a speed if 40mph. Instead of flying due north, the pilot sets her course at an angle of 50 degrees to the east of north. A strong wind is blowing directly from the south at a speed of 25mph.

a) Use a ruler and protractor to draw a vector parallelogram to scale.


North
!
!
!
!
!
!
!
____________!____________ (This is an example us how thw graph
! looks. )
!
!
^ !
! !
! !
! !
Wind

b) Draw the resultant vector.
c) Measure to determine her actual velocity and the angle of the her path from due north.

5. Tennis balls are often purchased in a cylindrical canister. Three tennis balls fit perfectly into a cylinder. Which is greater- the circumference of the canister of the height of the canister? You must support your answer with a picture and a written explanation.

CAN YOU PLEASE HELP!!!!!!!!!
THANK YOU SO MUCH!!!!!!!!!

 

[jolly]

That's a nice laundry list. Now what have you done on these problems so far?

 

[soroban]

Hello, Twins!

Here are the first two . . .

1. Consider a circle centered at the origin.
Point A on the circle has the coordinates (3,4).

a) What is the slope if the radius drawn to the point A?
b) What is the slope of the tangent line through point A?
c) Write an equation for the line tangent to the circle at point A?

                  |    \ 
                * * *   \
            *     |     *\   A
          *       |       o(3,4)
         *        |     /  *
                  |   /     \
        *         | /       *\
      - * - - - - + - - - - * \
        *         |O        *
                  |
         *        |        *
          *       |       *
            *     |     *
                * * *
                  |

(a) The slope of the line from \(\displaystyle (0,0)\) to \(\displaystyle (3,4)\) is: \(\displaystyle \,m\:=\:\frac{4\,-\,0}{3\,-\,0}\:=\:\frac{4}{3}\)


(b) A tangent to a circle is perpendicular to the radius at the point of tangency.
\(\displaystyle \;\;\)Hence, the slope of the tangent is: \(\displaystyle \,-\frac{3}{4}\)


(c) We have a point on the tangent \(\displaystyle (3,4)\) and its slope: \(\displaystyle \,m\,=\,-\frac{3}{4}\)
From the Point-Slope Formula: \(\displaystyle \,y\,-\4\:=\:-\frac{3}{4}(x\,-\,3)\)
\(\displaystyle \;\;\)which simplifies to: \(\displaystyle \,y\:=\:-\frac{3}{4}x\,+\,\frac{25}{4}\)

2. Circle O centered at the origin passes through points A(3,4) and point B(4,-3).
Write an equation for the perpendicular bisector of segment AB.
Sketch the line on the graph.

                  |
                * * *
            *     |     o A(3,4)
          *       |       *
         *        |        *
                  |
        *         |         *
      - * - - - - + - - - - *
        *         |         *
                  |
         *        |        o B(4,-3)
          *       |       *
            *     |     *
                * * *
                  |

The perpendicular bisector of AB contains the midpoint of AB.
\(\displaystyle \;\;\)Midpoint of AB \(\displaystyle \,=\:\left(\frac{3+4}{2},\,\frac{4-3}{2}\right)\:=\:\left(\frac{7}{2},\,\frac{1}{2}\right)\)

The perpendicular bisector of B is perpendicular to AB.
\(\displaystyle \;\;\)Slope of AB \(\displaystyle \,=\,\frac{-3-4}{4-3}\,=\,-7\)
\(\displaystyle \;\;\)Slope of the perpendicular bisector: \(\displaystyle \,m\,=\,\frac{1}{7}\)

We have a point \(\displaystyle \left(\frac{7}{2},\,\frac{1}{2}\right)\) and the slope \(\displaystyle m\,=\,\frac{1}{7}\)
Point-Slope Formua: \(\displaystyle \,y\,-\,\frac{1}{2} \;=\;\frac{1}{7}\left(x\,-\,\frac{7}{2}\right)\)
\(\displaystyle \;\;\)which simplifies to: \(\displaystyle \,y\:=\:\frac{1}{7}x\)

 

[soroban]

Hello, Twins!

The last one is easy . . . if you make a sketch . . .

5. Tennis balls are often purchased in a cylindrical canister.
Three tennis balls fit perfectly into a cylinder.
Which is greater: the circumference of the canister or the height of the canister?
You must support your answer with a picture and a written explanation.

      +---*-*---+
      | *     * |
      |*       *|
      |*       *|
      | *     * |
      |   * *   |
      | *     * |
      |*       *|
      |*       *|
      | *     * |
      |   * *   |
      | *     * |
      |*       *|
      |*       *|
      | *     * |
      +---*-*---+

Let \(\displaystyle r\) be the radius of the tennis ball.

The circumference of the canister is: \(\displaystyle \,2\pi r\)

The height of the canister is: \(\displaystyle \,6r\)

Since \(\displaystyle 2\pi\:>\:6\), the circumference is greater.