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#1 2014-07-30 04:53:17

harrychess
Member
Registered: 2014-04-04
Posts: 34

Medians and Diagonals of A Trapezoid

Please Help! I need someone to explain how to work this problem.

The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid.

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#2 2014-07-30 06:25:30

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 97,273

Re: Medians and Diagonals of A Trapezoid

Hi

I am getting


In mathematics, you don't understand things. You just get used to them.

If it ain't broke, fix it until it is.

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#3 2014-07-30 20:52:52

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,193

Re: Medians and Diagonals of A Trapezoid

hi harrychess,

By using Sketchpad and measuring, I'm getting this:

eBa4bHw.gif

AC = 10, BD = 8.  AB parallel to DB.

I can move E about, and still get the same result for FG, which is interesting as the whole shape changes.

Using those right angled triangles, it's not too hard to show that the area of ABCD = 0.5 x 8 x 10.

So if I could calculate h, the height of the trapezium, then the length of FG would be easy.

But I cannot see how to get h just yet.  Still thinking ....................................

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2014-07-31 00:47:31

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,193

Re: Medians and Diagonals of A Trapezoid

Ok. Extra bits on diagram:

ZAl5q7d.gif

h = HD = CI.  The angles marked with a dot are equal (alternate angles).  Say, alpha.

So using the area I have already worked out and

where a and b are the lengths of the parallels, you can work out FG.

Bob

Last edited by bob bundy (2014-07-31 00:49:14)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2015-08-11 06:45:02

Lenchen
Member
Registered: 2015-08-11
Posts: 2

Re: Medians and Diagonals of A Trapezoid

Can somebody explain the solution to this problem without using trig?

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#6 2015-08-11 08:40:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 97,273

Re: Medians and Diagonals of A Trapezoid

Of course trig is the best way, the thematic way, but if you must avoid it you could use the computer. Geogebra can solve it if you are not too concerned with rigor and just want the answer.

Here is some info on this:

http://www.mathsisfun.com/geometry/trapezoid.html


In mathematics, you don't understand things. You just get used to them.

If it ain't broke, fix it until it is.

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#7 2015-08-11 08:51:16

Lenchen
Member
Registered: 2015-08-11
Posts: 2

Re: Medians and Diagonals of A Trapezoid

I would like a way to prove it using just algebra and geometry

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#8 2015-08-11 08:54:08

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 97,273

Re: Medians and Diagonals of A Trapezoid

Did you make a drawing first?

This drawing covers the cases and it is easy to get the coordinates of each point. You want the distance of the red line.

UkbRWTa.png


In mathematics, you don't understand things. You just get used to them.

If it ain't broke, fix it until it is.

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#9 2015-08-11 20:34:26

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,193

Re: Medians and Diagonals of A Trapezoid

hi Lenchen,

Welcome to the forum.

Trig works because of the properties of similar triangles, so what you ask should be possible.  My answer is in three parts, with three diagrams.  To keep each part as simple as I can I will just show what is needed in each diagram.

Part one: calculate the area ABCD.

7L72dNU.gif

I have boxed in the shape.  This box has an area of 8 x 10.  The sides of the shape cut across the middle of four rectangles, splitting the box into 4 pairs of right angled triangles.  So the area of ABCD must be half the box, ie. 40.

Part two: 

bEp7CCW.gif

GF is perpendicular to AB.

triangles BEF and DEG are similar, so

Similarly

and

If DE = 8k and CE = 10k then CD = root(164)k  Sorry, couldn't avoid Pythagoras here.

Part 3.

iHDX7xI.gif

H is the foot of the perpendicular on AB.

angle DBH = EDC (by parallels)

Consider triangles BDH and DCE

They have one angle equal by above and one right angle so they are similar.

Therefore

The area of a trapezium is

where a and b are the lengths of the parallels and h the perpendicular distance between them.  Also the median will be the average of a and b  so

Bob

ps.  I have been asked to help with a lot of geometry recently, including the same question more than once.  I'm not complaining; I like to be helpful; but I was wondering whether it would be helpful to members if I created a post consisting of a geometry contents list with links to the relevant posts.  What do you think of this idea?

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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