Geometry High school level Help

[AAA]

Hi Guys,

I need help with this. Should I use the Pythagorean theorem for this problem?

 

[lex]

No. Similar triangles.

 

[Deleted member 4993]

Hi Guys,

I need help with this. Should I use the Pythagorean theorem for this problem?

Yes - if you want to - that will give you the length of LY.

Then use "similar triangles" to calculate the length of LW.

 

[lex]

Similar triangles: [MATH]\bigtriangleup \text{WAY, } \bigtriangleup \text{LAY}[/MATH](No need for Pythagoras).

 

[AAA]

Thank you guys!

 

[AAA]

Can you explain to me the steps I should take?

 

[HallsofIvy]

As you were told, triangle WLY and triangle YLA are similar triangles. Specifically the have angle L in common and each have a right angle. Therefore the third angles are the same. So theIr sides are in the same proportion.

We do need the Pythagorean theorem because the side whose length we are asked to find, LW, the hypotenuse of triangle WLY, corresponds to LY, the hypotenuse of YLA. We have \(\displaystyle \frac{LW}{LY}= \frac{LY}{AY}\).

You need the Pythagorean theorem to find LY,

 

[lookagain]

We have \(\displaystyle \frac{LW}{LY}= \frac{LY}{AY}\).

Please check that proportion against that diagram.

 

[Deleted member 4993]

Can you explain to me the steps I should take?

Can you list the similar triangles?

I'll start off:

triangle LAY is similar to LWY

triangle YAW is similar to LWY

Continue...

 

[lex]

Can you explain to me the steps I should take?

Let the angle at L be size 'x'
Fill in the sizes of the other angles.
(Look for similar triangles).
Swing triangle ALY onto triangle AWY.
...

Work out AW.
Hence, work out LW.

 

[AAA]

How do I know which triangles are similar? Thanks btw for the explanation.

Let the angle at L be size 'x'
Fill in the sizes of the other angles.
(Look for similar triangles).
Swing triangle ALY onto triangle AWY.
...

Work out AW.
Hence, work out LW.

View attachment 26662

 

[lex]

Triangles are similar when the three angles in one are the same as the three angles in the other (i.e. the set of three angles in one is the same as the set of three angles in the other).
That's why I said label the angle down at L, as xº. Then fill in all the others. So you are looking for triangles which contain this set of angles:
xº, 90º, and (90-x)º
One triangle is a 'scaled up' version of the other.

 

[pka]

I need help with this. Should I use the Pythagorean theorem for this problem?

In any right triangle the length of the altitude onto the hypotenuse is a mean proportional between the parts( proven with similar triangles).
So \(\dfrac{LA}{YA}=\dfrac{YA}{AW}\). From which \(LW=LA+AW\).

 

[lex]

In any right triangle the length of the altitude onto the hypotenuse is a mean proportional between the parts( proven with similar triangles).
So \(\dfrac{LA}{YA}=\dfrac{YA}{AW}\). From which \(LW=LA+AW\).

A handy little wrinkle that. I hadn't heard that expressed so.
Nice.