# Area of a right-angled triangle

1. Jan 27, 2004

### Chen

In a given right-angled triangle, the length of the hypotenuse is 10 and the length of one of the heights is 6. What is the area of the triangle?

a) 12
b) 30
c) 24
d) 24 or 30

Last edited: Jan 28, 2004
2. Jan 27, 2004

### chroot

Staff Emeritus
The area of any triangle is

$$\frac{1}{2} b h$$

where b is the length of the base of the triangle, and h is its height.

- Warren

3. Jan 27, 2004

### HallsofIvy

I always get a kick out of people who just quote a problem. Do we get points for a correct answer?

A (almost!*) always, Chroots answer is correct- the area of any triangle is (1/2)bh where b is the "base", one of the sides of the triangle and h is the "height", measured perpendicular to the base.

In a right triangle, the two legs are perpendicular so you can use one as base and the other as height.
Use the Pythagorean theorem to find the length of the other leg of the triangle, then apply "(1/2)bh".

* I, myself, never make mistenk(s!

4. Jan 28, 2004

### Chen

Ahh, but I do make mistakes. The problem should read "the length of one of the heights is 6". What would be the correct answer then? (Bearing in mind that the given height could be the height to the hypotenuse.)

5. Jan 28, 2004

### chroot

Staff Emeritus
The triangle has more than one height?

Listen, we've told you how to do this problem, but we're not going to do it for you. Try it, it's not hard.

- Warren

6. Jan 28, 2004

### Chen

I do know how to solve this problem and already have, years ago. I am just trying to see if you can, but clearly I overestimated you.

And yes, every triangle - the thing with three sides, you know - has three heights. Not just one. But three. (And it's this little factoid that makes the problem more than just a 1st grade exercise.)

Last edited: Jan 28, 2004
7. Jan 28, 2004

### chroot

Staff Emeritus
No, if you first select one side as a base, a triangle has only one height. Sorry.

- Warren

8. Jan 28, 2004

### Chen

Where have I selected a base? All the problem says is that the length of one of (the three) heights in this triangle is 6.

9. Jan 28, 2004

### chroot

Staff Emeritus
We don't really have time here for your little riddles. Sorry. At the very least, if you're "challenging" us with a riddle, label it as such. We thought you were a schoolchild who didn't understand how to find the area of triangles.

- Warren

10. Jan 28, 2004

### Integral

Staff Emeritus
Your initial statement completely specifies the triangle, where is the difficulty?

The triangle you describe is a 3-4-5 triangle, one of the most useful combinations of all right triangles. Now tell us why finding this area should be difficult?

11. Jan 28, 2004

### Chen

Is it not possible that the height whose length is 6, is perpendicular to the hypotenuse of this triangle?

12. Jan 28, 2004

### Chen

I did not realize that the answer to a question depends on the inquirer. And if you really thought I were a schoolchild you could at least pretend to be nice about it, because if I really were a 10 year old your replies would have been insulting to me. Luckily I have passed that age and I am just saddened by the fact that it is a mentor - who should set an example for others - that speaks like this.

Last edited: Jan 28, 2004
13. Jan 28, 2004

### matt grime

So you've got one those infinitely large triangles lying around have you?

14. Jan 28, 2004

### chroot

Staff Emeritus
Your question, as originally posted, was "what is the area of a right triangle with one leg 6 feet and hypotenuse 10 feet?", which is a schoolchild's question.
Oh no.

- Warren

15. Jan 28, 2004

### Integral

Staff Emeritus
Perhaps a link to a drawing to this "triangle" would clear up the questions. It is said a drawing is worth a 1000 words. Perhaps english is not your native tounge and you do not realize what you are saying.

16. Jan 29, 2004

### KingNothing

Lets calm down everyone. I think he means he knows the hypotenuse and one leg. If so, it's $$10^2=6^2+A^2$$. Therefore, $$100=36-A^2$$ and 64=A^2 and A=8. 8 is your other leg. 8*6=48, so your area is 24. It's pretty late here, so I hope I didnt skip anything, its been so long since I even touched a problem like this.

Please stop fighting, people. There's no reason to act negative towards others here.

That's a good theory, Integral. His profile says he is from Israel, and if it is true, it makes sense to me to say some things incorrectly.

Last edited: Jan 29, 2004
17. Jan 29, 2004

### Integral

Staff Emeritus
I don't think he is talking about side length, but rather perpendicular bisectors.

18. Jan 29, 2004

### lastlaugh

like the triangle is sitting on the hypotonuse???? is this a 45-45-90 triangle or can we not make that assumption?

19. Jan 29, 2004

### chroot

Staff Emeritus
Either 24 or 30. If the triangle is sitting on either of its legs, it's 24. If the triangle is sitting on its hypotenuse, it's 30.

- Warren

20. Feb 1, 2004

### oen_maclaude

i guess, everybody had his/her contribution. i hope that chen would have understood it in anyway.

However, if chen haven't pictured the situation, then there is still a need for us to draw the figure to help him/her out.

File size:
24.5 KB
Views:
135