the diagonal of a square is 2cm longer than a side, Find the length of a side

Question

It asks me to identify the variable, draw a picture, write the equation, solve the equation and to write the answer in English.

Answer 1

Xiomy, the variable is going to be the length of the diagonal, because whatever length you make the sides, the diagonal will have to be 2 cm longer.

 

Your picture starts as a square. Along each side, write "s" for side.

 

Now draw the diagonal between 2 opposite corners, and label it "s + 2."

 

As D.Y. noted, this is a right triangle, so you can use the Pythagorean theorem to set up the equation:

 

s² + s² = (s + 2)²

 

Or, 2s² = (s + 2)(s + 2), or

 

2s² = s² + 4s + 4

 

To solve it, you might have to use the quadratic formula, which is above my skill level. :-)

 

Eyeballing it, I'd say the sides are going to be 5 point-something, diagonal would be 7 point-the-same-something.

 

I know this doesn't constitute a full answer, but it shows that it can be done.

 

Hope this helps--even a little bit. :-)

 

Matt in New York

Answer 2

Hello, Xiomy

This is a 45-45-90. If one of the leg is x, then the hypotenuse must be x times the square root of two. So, let me find a leg. (Remember, this a square, so all the sides are congruent, thus the two legs are congruent) 

 

I am going to continue here Matt left off.  I am going to use the quadratic formula. So, Matt got

 

2s^2=s^2+4s+4

-s^2    -s^2

s^2= 4s+4

-4s.     -4s                 Since I am going to use the quadratic 

s^2-4s=4                  formula, I am moving all the terms to

        -4   -4               one side of the equation.

s^2-4s-4= 0              So, a=1, b=-4, c=-4

 

You might recall the quadratic formula.  I am going to do what is inside the radical sign, which is called the Determinant. The determinant can be found by using 

 

b^2-4ac                     I am sure you remember. Now, substitute

(-4)^2-(4)(1)(-4)        the values of a,b and c.

16-(-16)= 32              This 32 goes underneath the radical sign

                                  when we do the quadratic formula.

I hate typing it, since I am using an ipad, but to help you out, I am willing to endure the mental anguish....yes, the quadratic formula

 

[(-b)+/- (square root of b^2-4ac)]/2a.       Yes, I did it. We, 

[(-b)+/-(square root of 32)]/2a                  already figure out

                                                               whats underneath 

                                                               the radical

Next, I am going to substitute for a and b. So, we have

[-(-4)+/-(sq.rt. of 32)]/2(1)

[4+/-(sq.rt of 32)]/2                                 The sq.rt. of 32, can be 

[4+/-(4 times sq.rt of 2)/2                         simplified as sq.rt 16 

                                                                times sq.rt. 2 or 

                                                                4 times sq.rt.2

[2+/-(2 sq.rt 2]                                         Simplifying by 

                                                                dividing by 2.

 

Ok, we are almost there.. Now, we have two possible answers for x, the length of legs

 

x= 2+2sq.rt 2.        or  x= 2-2sq.rt. 2

                                   This can't be an

                                   answer. Legs can't be negative.

 

So, each leg is 2+2 times sq.rt 2. Since this is a 45-45-90 right triangle, we can find the hypotenuse by multiplying the leg by sq.rt.2. Let's do that, 

(2+2sq.rt.2)(sq.rt.2)= 2 sq.rt.2+4

Note that when you multiply sq.rt. 2 times sq.rt. 2, our answer is 2. So, we have 2*2=4.  That's where we got the 4 just in case you are wondering.  

 

There you have it Xiomy, about 7 hours late, and after about 30 minutes of typing on my I pad with my index finger..... drums please, the answer.

 

The legs are each (2+2 sq.rt. 2) and the hypotenuse is 2 sq.rt.2+4.

Finally, my finger hurts. :) good luck.. Don't forget to rate my answer.

 

D. Y.