# Surd problem

• Mar 12th 2009, 02:06 AM
sammy28
Surd problem
hi all,

this is a problem that requires the answer expressed as a surd. (calculator not to be used)

The sides of a rectangle are in the ratio 2:3. The diagonal is of length 26cm. Find the perimeter.

The answer given in the answer section is $\displaystyle 20\sqrt13$

$\displaystyle 26^2 = 626$

sides are therefore

$\displaystyle \sqrt{\frac{2}{5} * 626} \mbox{ and} \sqrt{\frac{3}{5} * 626}$

and perimeter

$\displaystyle 2\sqrt{\frac{1352}{5}} + 2\sqrt{\frac{2028}{5}}$

using prime factors of 1352 and 2028

$\displaystyle 26\sqrt{\frac{8}{5}} + 52\sqrt{\frac{3}{5}}$

using a calculator the answer i calculated is not equal to the textbooks answer
• Mar 12th 2009, 02:17 AM
Quote:

Originally Posted by sammy28
hi all,

this is a problem that requires the answer expressed as a surd. (calculator not to be used)

The sides of a rectangle are in the ratio 2:3. The diagonal is of length 26cm. Find the perimeter.

The answer given in the answer section is $\displaystyle 20\sqrt13$

$\displaystyle 26^2 = 626$

sides are therefore

$\displaystyle \sqrt{\frac{2}{5} * 626} \mbox{ and} \sqrt{\frac{3}{5} * 626}$

and perimeter

$\displaystyle 2\sqrt{\frac{1352}{5}} + 2\sqrt{\frac{2028}{5}}$

using prime factors of 1352 and 2028

$\displaystyle 26\sqrt{\frac{8}{5}} + 52\sqrt{\frac{3}{5}}$

using a calculator the answer i calculated is not equal to the textbooks answer

First of all 26^2 =676

I dont know what you are trying to do

Lets say the sides are 2x & 3x

Hence
$\displaystyle Diagonal = \sqrt{4x^2+9x^2} = 26$

$\displaystyle x\sqrt{13} = 26$

Divide both sides by squareroot(13)

$\displaystyle x = 2\sqrt{13}$

Perimeter $\displaystyle = 2(2x + 3x) = 10 x = 20\sqrt{13}$
• Mar 12th 2009, 02:45 AM
sammy28

i didnt think about using algebra, i was just taking the ratio as a total 676=5/5 and expected the result to be the same. I need to figure out where i went wrong (Giggle)
• Mar 12th 2009, 06:25 AM
sammy28
i played with this abit more im still confused as to why the above not work but using adarsh algebra

$\displaystyle 26 = \sqrt{(\frac{2}{5}x)^2 + (\frac{3}{5}x)^2}$

$\displaystyle 26 = \sqrt{\frac{4}{25}x^2 + \frac{9}{25}x^2}$

$\displaystyle 26=\sqrt{\frac{13}{25}x^2} \equiv x\sqrt{\frac{13}{25}}$

$\displaystyle x = 26 \div \sqrt{\frac{13}{25}} \equiv \frac{26 * 5}{\sqrt{13}} \equiv \frac{13 * 10}{13^\frac{1}{2}} \equiv 10\sqrt{13}$

perimeter 2 x sides

$\displaystyle 2 \mbox{x} 10\sqrt{13} \equiv 20\sqrt{13}$
• Mar 12th 2009, 11:32 AM
Quote:

Originally Posted by sammy28
i played with this abit more im still confused as to why the above not work but using adarsh algebra

$\displaystyle 26 = \sqrt{(\frac{2}{5}x)^2 + (\frac{3}{5}x)^2}$

$\displaystyle 26 = \sqrt{\frac{4}{25}x^2 + \frac{9}{25}x^2}$

$\displaystyle 26=\sqrt{\frac{13}{25}x^2} \equiv x\sqrt{\frac{13}{25}}$

$\displaystyle x = 26 \div \sqrt{\frac{13}{25}} \equiv \frac{26 * 5}{\sqrt{13}} \equiv \frac{13 * 10}{13^\frac{1}{2}} \equiv 10\sqrt{13}$

perimeter 2 x sides

$\displaystyle 2 \mbox{x} 10\sqrt{13} \equiv 20\sqrt{13}$

I think you got it But just as an assurance I will do it your way

Lets consider the sides to be 3x/5 and 2x/5

Now we follow the steps (same as you did here)

$\displaystyle 26 = \sqrt{(\frac{2}{5}x)^2 + (\frac{3}{5}x)^2}$ ....this thing came from Pythagoras theorem

$\displaystyle 26 = \sqrt{\frac{4}{25}x^2 + \frac{9}{25}x^2}$

$\displaystyle 26=\sqrt{\frac{13}{25}x^2} \equiv x\sqrt{\frac{13}{25}}$

$\displaystyle x = 26 \div \sqrt{\frac{13}{25}} \equiv \frac{26 * 5}{\sqrt{13}} \equiv \frac{13 * 10}{13^\frac{1}{2}} \equiv 10\sqrt{13}$

Perimeter is the sum of lengths of all sides

Thus what we basically did was

(3x/5 + 2x/ 5) + (3x/5 + 2x/5) = x + x = 2x =Answer
• Mar 13th 2009, 01:16 AM
sammy28
thank you for explaining it so thoroughly, my problem solving skill is not as good as it should be. I need to think before jumping into questions (Hi)