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### Scaling a Right Triangle

```Date: 12/01/2003 at 21:05:12
From: RedMan
Subject: Right Triangle

A right triangle has sides of length a = 30 ft, b = 8 ft, and
c = 31.0483 ft.

Next, side c is move inward 4 ft, while angles A, B, and C remain the
same.

How do I determine the new lengths of a, b, and c?
```

```
Date: 12/02/2003 at 20:29:44
From: Doctor Douglas
Subject: Re: Right Triangle

Hi RedMan.

Thanks for writing to the Math Forum.

Here is a diagram of the triangle:

P--------------q-------Q         PQ = a = 30 ft
|              . y               PR = b = 8 ft
r      .                         RQ = c = 31.0483 ft
R

I assume you mean that line c is moved towards P by 4 ft, leading to
new triangle Pqr.

Consider point y, which lies on line RQ and is moved 4 ft inward to
new point q on line PQ.  qyQ is a right triangle, so

yq = 4 ft = (qQ)*sin(Q)

Thus, the top side PQ is shortened to become Pq, i.e., it becomes

Pq = PQ - qQ

= PQ - [4 ft / sin(Q)]

Thus, each side of the original triangle is shrunk by multiplication
by the following factor

Pq         (4 ft)                 4
-- = 1 - -----------  = 1 - -------------- = 0.4825
PQ       (PQ)*sin(Q)        30*(8/31.0483)

Thus,

Pq = 0.4825*(30 ft)      = 14.47 ft
Pr = 0.4825*(8 ft)       =  3.86 ft
rq = 0.4825*(31.0483 ft) = 14.98 ft

Note that these sides still form a right triangle Pqr, and that it has
the same angles (3.86/14.47) = (8/30).

I hope that wasn't too complicated.

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

```
Date: 12/11/2003 at 22:04:32
From: RedMan
Subject: Thank you (Right Triangle)

Thank you!

Previously, this was performed by hand: first draw to scale, then
calculate and count certain multiple objects. You've made my job much
easier.

It's good to have a flashlight for a dark path, but much better when
someone shows you the light switch.

Thanks ever so much. I'm sure to have another question, in the future.

RedMan
```
Associated Topics:
High School Triangles and Other Polygons
High School Trigonometry

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