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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

>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.

- 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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