It is a basic geometry rule that you should memorize for the test. It will save you a bunch of time

It is the Pythagorean triplets or triples. Here more https://www.mathwarehouse.com/geometry/ ... iplets.php

One side is 5, one is 3 and the angle is 90°, the triangle must be a right triangle. So the other side is 4

Regards

Thank you, but I still don't understand how we can get the height of the other triangle (XYZ) ? The left triangle does not have a right angle so we can't apply any of the triplet rules... Why is its height 4 and why is the base 6? Shouldn't the base be the longest side? (the unknown side)?
I understand how to get the height of the triangle XYW but that is not being asked. The question is asking for the area of: XYZ.
I seem to be stuck..

No, you are not stuck. You are right and the question is flawed. At most it should ask for the area of XYW
If we wanna the area of XYZ we can calculate the entire area of the triangle XYW which is \(\frac{b \times h}{2} = \frac{9 \times 4}{2} = 18\)

Then the area of YZW which is \(\frac{3 \times 4}{2}= 6\)

\(XYW - YZW = 12\)

What you pointed out is totally correct, at least in my view.

I believe Jacqueline may be confusing the length of the side of the triangle and it's height, which are two different things. (It's a common mistake).

Imagine the that leaning triangle is a building and someone asks you for the height of that building. What are they really asking you? Are they asking you how long the side of the building is? No. They are asking you the distance from the tip of that building to the ground.
Better yet. Pretend they ask you to drop something from the top of that building. What path will it follow? Will it follow a path down the side of the triangle? No. It will follow a path straight down. That is your height. That path straight down is the height of the triangle. That path is the dotted line from Y to W.

If you want to know the height of any triangle draw a line extending in both directions from the base. Draw a parallel line to the base that just touches the opposing vertex of the triangle. That distance between those two parallel lines, which is the length of a perpendicular line between them, is the height of your triangle.

It might seem like they are making a mistake in asking for the area of XYZ, but it is not. The question is designed to make sure one knows that the height of a triangle is not the same as the length of it's side.

Another thing, as for how we know the base of XZ is 6. It was given.

Here is another problem that implicitly covers the same subject: (https://gre.myprepclub.com/forum/in-the-fi ... tml#p38039).

Start by redrawing the figure, filling in all the information given in the text. To find the area of triangle XYZ, a base and a height are required. If side XZ is a base, then YW can act as a height, as these two are perpendicular.
Because triangle ZYW is a right triangle with two known sides, the third can
be determined using the Pythagorean theorem: ZY is the hypotenuse, so (a)2 +(3)2 = (5)2
, meaning that a2 + 9 = 25 and a2 = 16, so a = 4.
Alternatively, recognize the Pythagorean triple: ZYW is a 3–4–5 triangle:
The area of triangle XYZ is b*h/2=6*4/2=12

YZ = 5
ZW = 3
Angle W = 90

By applying the Pythagoras theorem,

YZ^2 = YW^2 + ZW^2
=> YW^2 = YZ^2 - ZW^2 = 25 - 9 = 16
=> YW = 4

So, the area of the triangle XYZ = the area of the triangle XYW - the area of the triangle ZYW = .5*(6+3)*4 - .5*3*4 = 12