# Thread: Measure of AC

1. ## 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

2. ## 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?

3. ## 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

4. ## Re: Measure of AC

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.

5. ## Re: Measure of AC

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

6. ## Re: Measure of AC

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.

7. ## Re: Measure of AC

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.

8. ## Re: Measure of AC

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:

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

$\bar{AD}=u-1$

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

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

By Pythagoras, we have:

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

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

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

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

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

The only root that satisfies all of the criteria is:

$u=\bar{AC}=5$