Pythagoras Theorem: If XY = 10 cm and YZ = 2XZ calculate the lengths of XZ and YZ

[JimCrown]

The question I am having trouble with is:

Triangle XYZ has:

X = 90 degrees

If XY = 10 cm and YZ = 2XZ calculate the lengths of XZ and YZ

I have tried to do the Pythagoras theorem:

a^2 = b^2 + c^2

2XZ^2 = 10^2 (XY) + YZ^2


I changed the XZ and YZ just to X

And tried to work it out like:

2x^2 = 10^2 + x^2

But after subtracting X it didn't work out the length of the hypotenuse correctly. As I got x^2 = 100 which means X is 10 but 10^2 + 10^2. Then square rooting that did not get the hypotenuse (2XZ) double the length of YZ.

Can anybody help me please?

 

[JeffM]

The question I am having trouble with is:

Triangle XYZ has:

X = 90 degrees

If XY = 10 cm and YZ = 2XZ calculate the lengths of XZ and YZ

I have tried to do the Pythagoras theorem:

a^2 = b^2 + c^2

2XZ^2 = 10^2 (XY) + YZ^2


I changed the XZ and YZ just to X

And tried to work it out like:

2x^2 = 10^2 + x^2

But after subtracting X it didn't work out the length of the hypotenuse correctly. As I got x^2 = 100 which means X is 10 but 10^2 + 10^2. Then square rooting that did not get the hypotenuse (2XZ) double the length of YZ.

Can anybody help me please?

Name things.

\(\displaystyle \text {Length of XZ } = a. \\

\text {Length of XY } = b.\\

\text {Length of YZ } = c.\)

So three unknowns. That means you need to find three relationships. But that is easy.

\(\displaystyle b = 10.\\

c = 2a.\\

a^2 + b^2 = c^2 \implies a^2 + 10^2 = (2a)^2 \implies \\

a^2 + 100 = 4a^2 \implies 3a^2 = 100 \implies 9a^2 = 300 \implies \\

\sqrt{9a^2} = \sqrt{300}= \sqrt{100 * 3} \implies 3a = 10 \sqrt{3} \implies \\

a = \dfrac{10\sqrt{3}}{3} \implies 2a = \dfrac{20\sqrt{3}}{3} = c.\).

Now check.

\(\displaystyle \left (\dfrac{10\sqrt{3}}{3} \right )^2 + 10^2 = \dfrac{100 * 3}{9} + 100 = \dfrac{300}{9} + \dfrac{900}{9} = \dfrac{1200}{9}.\)

\(\displaystyle \left ( \dfrac{20\sqrt{3}}{3} \right )^2 = \dfrac{400 * 3}{9} = \dfrac{1200}{9}.\)

You were very much on the right track, but make the notation work for you.

 

[sinx]

The question I am having trouble with is:

Triangle XYZ has:

X = 90 degrees

If XY = 10 cm and YZ = 2XZ calculate the lengths of XZ and YZ

I have tried to do the Pythagoras theorem:

a^2 = b^2 + c^2

2XZ^2 = 10^2 (XY) + YZ^2


I changed the XZ and YZ just to X

And tried to work it out like:

2x^2 = 10^2 + x^2

But after subtracting X it didn't work out the length of the hypotenuse correctly. As I got x^2 = 100 which means X is 10 but 10^2 + 10^2. Then square rooting that did not get the hypotenuse (2XZ) double the length of YZ.

Can anybody help me please?

2x^2 = 10^2 + x^2
you neglected to square the 2,
it should be 4x^2 = 10^2 + x^2