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Parallelogram Side Length

Date: 08/12/2003 at 02:49:27
From: Jason
Subject: Angles and shapes

In a parallelogram ABCD, K is a point on AB such that angle DKC = 90 
degrees and AD = AK. If AD = 10cm, find the length of AB.

I tried using the Pythagorean theorem but I think that it won't work.

Date: 08/13/2003 at 13:14:50
From: Doctor Edwin
Subject: Re: Angles and shapes

Hi, Jason.

It took me a while to find a reasonable answer. We don't know much in 
terms of actual length, do we? That makes figuring out what to do a 
little easier. If they gave you a bunch of lengths you would have to 
choose from a lot of different strategies. As it is, we only have one 
length, and we have to use it to determine another.

What we need to do is find some way to relate the length of AD (or 
AK) to the length of KB or DC. Stop a minute and make sure that makes 
sense to you. In order to do it we're going to use some knowledge 
about angles and parallel lines.

Draw your parallelogram with the point K and the right triangle DKC. 
Label AD and AK as 10cm. Since opposite sides of a parallelogram are 
congruent, you can also label BC as 10.

If DA = AK, then triangle DAK is isoceles, which means angle ADK = 
angle AKD. Let's give that angle a name, theta. Label each of those 
two angles with a little theta.

Since AB is parallel to DC, angle AKD must be congruent to angle KDC 
(do you see why? If not, write back). So write a theta in there as 

You know that the interior angles of a triangle add up to 180 degrees, 
right? So if we know two angles of a triangle we can find the third. 
Triangle DKC is a right triangle. DKC is a right angle, CDK is equal 
to theta, and so 

 angle KCD = 180 - 90 - theta
           = 90 - theta

Again because AB is parallel to DC, CKB must be congruent to KCD. So 
label both those angles 90 - theta. 

We know that angle ABC is congruent to angle CDA. We also know that 
CDA is equal to 2 * theta.

If angle ABC is equal to 2 * theta, what's angle BCK? There are a 
couple of ways to get the answer. Now, looking at the angles in 
triangle KBC, what can you tell about the relation between BC and BK?

Does this help? Please write back if you're still stuck.

- Doctor Edwin, The Math Forum 

Date: 08/13/2003 at 16:11:42
From: Jason
Subject: Angles and shapes

I get what you said, but is there a definite answer (like AB=10cm) 
or is there an answer like AB=theta * 4?

Thank you.

Date: 08/13/2003 at 16:15:46
From: Doctor Edwin
Subject: Re: Angles and shapes

There is a definite numeric answer. Go through the entire answer and 
see if you can work out the end.

- Doctor Edwin, The Math Forum 
Associated Topics:
High School Triangles and Other Polygons

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