How can we prove that a triangle is a right triangle ?

[RB574]

Hi,

In a DS problem, what information is sufficient to prove that a triangle with 3 sides x, y and z is a right triangle ?

- If we can somehow prove that one of the angles is a right angle, then we can definitely conclude that the triangle is a right triangle.
- I don't understand the way to prove a triangle is a right triangle by just having the lengths of the sides.
For example, if we know that x = 6, y = 8 and z = 10, then does that mean that the triangle is a right triangle ?
What if we are only given that x = 3 and y = 4 ?
Also if sum of the squares of any two sides is 25, then does that mean that the 3rd side is 5 and hence the triangle is a right triangle ?

Please advice.

[tim]

There are several ways to prove that a triangle is a right triangle:

1) ANGLES ONLY - If, for instance, you know that two angles add up to 90 degrees (or add up to the third angle), the other one must also.

2) SIDES ONLY - You have a right triangle if and only if the sum of the squares of the two smaller sides equals the square of the largest side. I think this was what you were asking, and to address a question you seem to be bringing up, there is no way to tell whether a triangle is a right triangle given zero angles and only two sides.

3) COMBINATION OF SIDES AND ANGLES - This is a little more tricky, but one example would be if you have two sides in a 2:1 ratio with a 60 degree angle between them, then you have a 30-60-90 triangle. I suspect any problem of this type on the GMAT would likely involve either a 30-60-90 triangle or a 45-45-90 triangle.

[RonPurewal]

in case you're really bad with math statements given in words (like me), tim's #2 is just the familiar pythagorean rule.

i.e., if a^2 + b^2 = c^2 (where 'c' is the longest side) then you know you're dealing with a right triangle.

[RonPurewal]

What if we are only given that x = 3 and y = 4 ?

now THIS ^^ is not the kind of thing you should have to ask on a forum. this is the type of thing you should be able to figure out by yourself.

• take two sticks, or pens, or pencils, with lengths in the ratio 3:4. (if you don't have these, just cut out two strips of paper, 3 and 4 inches long respectively.)

• INVESTIGATE the potential triangles you can make with those 'sticks'.

• you'll very quickly see that...
...there are a whole lot of possible triangles,
...the third side can be almost as small as 1 (if you make a really tiny angle with the two sticks),
...the third side can be almost as long as 7 (if you point the sticks almost directly away from each other).

[RonPurewal]

my point, above, is that you need to have an INVESTIGATIVE MENTALITY to do well in quant. basically, you have to just try things and see what happens.

yes, there are certain things that you would be well served by memorizing.
e.g., the above fact pointed out by tim—the a^2 + b^2 = c^2 thing—is not the kind of thing you can divine just by looking at cases.
...but there are very, very, VERY few such things. the vast majority of 'principles' and/or 'patterns' on the GMAT will be things you can readily discover just by investigating a few cases.

this, after all, is THE ENTIRE POINT OF THIS EXAM: it's not a test of knowing things. it's a test of figuring things out.