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Ratios, Geometry, Trigonometry


Date: 06/10/99 at 22:02:28
From: Cathy
Subject: Homeschool - Proportions and Trigonometry

Hi Dr. Math,

This is the homeschool teacher again. I have turned to you all for 
help in the past when I have run into trouble in teaching various math 
procedures. You have always helped in answering questions and in just 
boosting confidence levels here. I turn to you again for assistance. 

I am pretty sure I have answered some of these questions correctly but 
numbers 1 and 2 have me confused. I believe I answered number 2 
correctly, but I'm not sure.

1.  Three angles are in the ratio of 3:4:5. Find the first and third
    angles if they are supplementary.

2.  Three angles are in the ratio 3:4:5. Find the angles if they are
    three angles of a triangle.

    The answer (which I originally noted for no. 1): <1 = 45, <2 = 60,
    <3 = 75.

3.  In triangle ABC, segment DE is parallel to segment AC, segment AD
    equals 20 in., segment BD = 10 in., and segment BC = 40 in. Find
    segment BE.

    I came up with 13 and 1/3 in. (?)

4.  Chords AB and CD intersect at E. Segment CD is perpendicular to
    segment AB. "O" is a point (presumably the center) on segment CD.
    Segment OD = 16 and segment DE = 20.  Find x.

    In my diagram, x appears to be segment AE, but it could be
    segment AB. Assuming it is AE, I got a number x = 15.49, but
    this doesn't sound right.

5.  A meter stick casts a shadow 12 cm long. To the nearest degree,
    at what angle with the ground are the sun's rays shining?

    My diagram shows a right triangle. I can't figure it out with
    only one length and the 90 degree angle.

6.  A flagpole rests on the edge of the roof of a building 200 feet
    in height. Determine the height of the flagpole, to the nearest
    foot.

    I have a diagram showing the flagpole leaning upright against the
    side of the building at a right angle to the ground. At a ground
    distance of 300 ft. from the building, I have the diagonal rising
    up to the top of the building. Then there is another diagonal
    rising from the same ground spot to the top of the flagpole with
    a 7-degree increase from the first diagonal. I came up with
    261 ft. as an answer to the question. I hope you will be able to
    understand the diagram.  

7.  The radius of circle O equals 8. Segment PA is a tangent drawn
    from point P and equals 6. PBC is a secant and segment BC is a 
    diameter. Find segment PB. Does PB = 2?

As always, I thank you in advance for your help. It is reassuring to 
know there is a place this homeschool teacher can turn to when in 
doubt.

Cathy


Date: 06/11/99 at 12:44:46
From: Doctor Peterson
Subject: Re: Homeschool - Proportions and Trigonometry

>1.  Three angles are in the ratio of 3:4:5. Find the first and third
>    angles if they are supplementary.

I'd start by calling the angles 3x and 5x, and then write an 
equation to say they are supplementary:

    3x + 5x = 180

Now solve the equation.


>2.  Three angles are in the ratio 3:4:5. Find the angles if they are
>    three angles of a triangle.
>
>    The answer: <1 = 45, <2 = 60, <3 = 75.

This is almost the same question, since the angles in a triangle add 
up to 180 degrees, and your answers are right.


>3.  In triangle ABC, segment DE is parallel to segment AC, segment AD
>    equals 20 in., segment BD = 10 in., and segment BC = 40 in. Find
>    segment BE.
>
>    I came up with 13 and 1/3 in. (?)

Sounds right. The ratio of BD:BA is the same as BE:BC, so 
BE is 1/3 of BC.


>4.  Chords AB and CD intersect at E. Segment CD is perpendicular to
>    segment AB. "O" is a point (presumably the center) on segment CD.
>    Segment OD = 16 and segment DE = 20.  Find x.
>
>    In my diagram, x appears to be segment AE, but it could be
>    segment AB. Assuming it is AE, I got a number x = 15.49, but
>    this doesn't sound right.

Sounds right to me. I presume you meant O is the center (don't you 
wish they'd state everything clearly rather than make you assume?), so 
16 is the radius, which is the hypotenuse of the right triangle AEO.

Your care in pointing out what you're assuming from the picture means 
there's a mathematician hiding inside you. If only textbooks would be 
that careful!


>5.  A meter stick casts a shadow 12 cm long. To the nearest degree,
>    at what angle with the ground are the sun's rays shining?
>
>    My diagram shows a right triangle. I can't figure it out with
>    only one length and the 90 degree angle.

Actually, you have two lengths, since a meter stick is 100 cm long. I 
assume you're doing a bit of trigonometry; the angle A has tangent of 
100/12.

        +
        |\
        | \
        |  \
        |   \
        |    \
        |     \
     100|      \
        |       \
        |        \
        |         \
        |          \
        |           \
        |_           \
        | |          A\
        +--------------+
               12


>6.  A flagpole rests on the edge of the roof of a building 200 feet
>    in height. Determine the height of the flagpole, to the nearest
>    foot.
>
>    I have a diagram showing the flagpole leaning upright against the
>    side of the building at a right angle to the ground. At a ground
>    distance of 300 ft. from the building, I have the diagonal rising
>    up to the top of the building. Then there is another diagonal
>    rising from the same ground spot to the top of the flagpole with
>    a 7 degree increase from the first diagonal. I came up with
>    261 ft. as an answer to the question. I hope you will be able to
>    understand the diagram.  

Here's how I understand the description:

                         +
                        /|
                       / |x
                      /  |
                     /   |
                    /    *--------+
                   /    /|        |
                  /   /  |        |
                 /   /   |        |
                /   /    |        |
               /  /      |        |
              /  /    200|        |
             / /         |        |
            /7/          |        |
           //            |        |
          //             |        |
         //A    300      |        |
    ----*----------------+--------+--------

    A = arctan(200/300) = 33.69 degrees

    A + 7 = arctan((200+x)/300)

                 200+x
    tan(40.69) = -----
                  300

Solve this for x and you get 57.95 feet for the flagpole.


>7.  The radius of circle O equals 8. Segment PA is a tangent drawn
>    from point P and equals 6. PBC is a secant and segment BC is a 
>    diameter. Find segment PB. Does PB = 2?

Here's my picture:

                ***********
             ***           ***
          ***                 ***
         *                       *
       **                         **
      *                             *
      *                             *
     *             O                 *
     *               +               *
     *               |  \   8        *
      *              |     \        *
      *              |         \    *
       **           8|            + B?
         *           |           *   \
          ***        |        ***       \
             ****    |    ****              \
                 ****+****---------------------+
                     A            6            P

PB is 8 less than the hypotenuse, which is 10, so you're right.

Glad to be of service.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   


Date: 06/13/99 at 15:28:05
From: Cathy
Subject: Re: Homeschool - Proportions and Trigonometry

Thanks Dr. Peterson!

When I looked at your diagram of the flagpole resting on the edge of 
building, a light went off in my head. It dawned on me that the 
flagpole sat atop the building's edge and not parallel to the entire 
height of the building.

Thank you so very much for all of your help, you (and Dr. Math) are 
greatly appreciated!

Thanks,
Cathy


Date: 06/14/99 at 09:09:20
From: Doctor Peterson
Subject: Re: Homeschool - Proportions and Trigonometry

Hi, Cathy.

>When I looked at your diagram of the flagpole resting on the edge of 
>building, a light went off in my head! It dawned on me that the 
>flagpole sat atop the building's edge and not parallel to the entire 
>height of the building!

I got the impression you might have things a little inside-out. 
Sometimes interpreting the English is the hardest part of these 
problems!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Conic Sections/Circles
High School Euclidean/Plane Geometry
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons
High School Trigonometry

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