# Measure of AC

• Jan 5th 2013, 06:47 PM
rcs
Measure of AC
ABCD is rectangle. If AD = x + 1, CD= 2x -3 & AC= x+2, find AC.

MY ATTEMPT:
i used pythagorean theorems:
(x +1)^2+ (2x-3)^2 = (x+2)^2

i got the problem from our textbook ... And the answer without solution is at the back of the portion of the book. It says that the answer is 5.
how did they make it to 5, please let me understand.

thanks
• Jan 5th 2013, 07:03 PM
MarkFL
Re: Measure of AC
The Pythagorean theorem is a good place to begin as it relates the 3 lengths given...now what do you get when you expand the binomials?
• Jan 5th 2013, 08:35 PM
ibdutt
Re: Measure of AC
(AD)2 + ( CD)2 = (AC)2
(x+1)2 + ( 2x-3)2 = ( x + 2 )2
x^2 + 2x + 1 + 4x^2 – 12x + 9 = x^2 + 4x + 4
4x^2 -14x + 6 = 0
2x^2 -7x + 3 = 0
2x^2 - x – 6x + 3 = 0
( 2x – 1 ) ( x -3)= 0
X = ½ or x = 3. X cannot be ½ because then CD will become negative, thus discarding this value.
Now x = 3 thus AC = x + 2 = 3+2=5
• Jan 5th 2013, 09:11 PM
MarkFL
Re: Measure of AC
Quote:

Originally Posted by ibdutt
(AD)2 + ( CD)2 = (AC)2
(x+1)2 + ( 2x-3)2 = ( x + 2 )2
x^2 + 2x + 1 + 4x^2 – 12x + 9 = x^2 + 4x + 4
4x^2 -14x + 6 = 0
2x^2 -7x + 3 = 0
2x^2 - x – 6x + 3 = 0
( 2x – 1 ) ( x -3)= 0
X = ½ or x = 3. X cannot be ½ because then CD will become negative, thus discarding this value.
Now x = 3 thus AC = x + 2 = 3+2=5

You really should let the OP have a chance (say at least 24 hours) to respond to suggestions about how to work the problem on their own.
• Jan 6th 2013, 05:16 AM
rcs
Re: Measure of AC
Quote:

Originally Posted by MarkFL2
You really should let the OP have a chance (say at least 24 hours) to respond to suggestions about how to work the problem on their own.

why wait for the next 24 hours when it could be done today
• Jan 6th 2013, 05:18 AM
skeeter
Re: Measure of AC
Quote:

Originally Posted by rcs
why wait for the next 24 hours when it could be done today

... because you are the one that needs to work it out, not someone else.
• Jan 6th 2013, 08:33 AM
MarkFL
Re: Measure of AC
Quote:

Originally Posted by rcs
why wait for the next 24 hours when it could be done today

Good forum etiquette dictates that one not give a solution immediately after someone else has given a suggestion on how to proceed. This devalues the preceding post.

You would have learned more had you expanded the binomials yourself, and realized only 1 of the roots made sense in the context of the problem.
• Jan 6th 2013, 08:44 AM
MarkFL
Re: Measure of AC
Quote:

Originally Posted by rcs
ABCD is rectangle. If AD = x + 1, CD= 2x -3 & AC= x+2, find AC.

MY ATTEMPT:
i used pythagorean theorems:
(x +1)^2+ (2x-3)^2 = (x+2)^2

i got the problem from our textbook ... And the answer without solution is at the back of the portion of the book. It says that the answer is 5.
how did they make it to 5, please let me understand.

thanks

Another way to proceed would be to let:

$\displaystyle \bar{AC}=u$ and so:

$\displaystyle \bar{AD}=u-1$

$\displaystyle \bar{CD}=2u-7$

We see we require $\displaystyle u>\frac{7}{2}$.

By Pythagoras, we have:

$\displaystyle (u-1)^2+(2u-7)^2=u^2$

$\displaystyle u^2-2u+1+4u^2-28u+49=u^2$

$\displaystyle 4u^2-30u+50=0$

$\displaystyle 2u^2-15u+25=0$

$\displaystyle (2u-5)(u-5)=0$

The only root that satisfies all of the criteria is:

$\displaystyle u=\bar{AC}=5$