Probability of a triangle

**malaygoel** · June 19th 2006, 09:16 PM

Prove that the probability that two positive numbers, x and y both less than 1, written down at random with unity, yield a triplet(x,y,1) whish are the sides of an obtuse angled triangle is $\frac{\pi - 2}{4}$

Keep Smiling
Malay

**ThePerfectHacker** · June 20th 2006, 08:32 AM

I do not understand what you mean by a triplet?

**CaptainBlack** · June 20th 2006, 09:41 AM

Originally Posted by ThePerfectHacker

I do not understand what you mean by a triplet?

A set of three numbers that are the lengths of the sides of some triangle,

(or one of my chidren

)

RonL

**CaptainBlack** · June 20th 2006, 09:43 AM

Originally Posted by ThePerfectHacker

I do not understand what you mean by a triplet?

Have you considered revising you signature to :

1) do not make posts of cardinality greater than 1
2) do not discuss illegal operations on the forum
3) do not use improper integrals
4) do not abuse notation
5) Axiom of Choice is allowed.

RonL

**JakeD** · June 20th 2006, 11:49 AM

Originally Posted by malaygoel

Prove that the probability that two positive numbers, x and y both less than 1, written down at random with unity, yield a triplet(x,y,1) whish are the sides of an obtuse angled triangle is $\frac{\pi - 2}{4}$

Keep Smiling
Malay

Here's the picture. The probability is the area between the thicker lines.

$\setlength{\unitlength}{2.5cm} \begin{picture}(1,1) \qbezier(0,0)(0,0)(1,0) \qbezier(0,0)(0,0)(0,1) \qbezier(1,0)(1,0)(1,1) \qbezier(0,1)(0,1)(1,1) \linethickness{1.3pt} \qbezier(1,0)(1,0)(0,1) \qbezier(1.0, 0.0)(1.0, 0.41)(0.71, 0.71) \qbezier(0.71, 0.71)(0.41, 1.0)(0.0, 1.0) \end{picture} $

**MathGuru** · June 20th 2006, 12:04 PM

Originally Posted by JakeD

Here's the picture. The probability is the area between the thicker lines.

$\setlength{\unitlength}{2.5cm} \begin{picture}(1,1) \qbezier(0,0)(0,0)(1,0) \qbezier(0,0)(0,0)(0,1) \qbezier(1,0)(1,0)(1,1) \qbezier(0,1)(0,1)(1,1) \linethickness{1.3pt} \qbezier(1,0)(1,0)(0,1) \qbezier(1.0, 0.0)(1.0, 0.41)(0.71, 0.71) \qbezier(0.71, 0.71)(0.41, 1.0)(0.0, 1.0) \end{picture} $

wow! +rep

**malaygoel** · June 20th 2006, 06:42 PM

Originally Posted by JakeD

Here's the picture. The probability is the area between the thicker lines.

$\setlength{\unitlength}{2.5cm} \begin{picture}(1,1) \qbezier(0,0)(0,0)(1,0) \qbezier(0,0)(0,0)(0,1) \qbezier(1,0)(1,0)(1,1) \qbezier(0,1)(0,1)(1,1) \linethickness{1.3pt} \qbezier(1,0)(1,0)(0,1) \qbezier(1.0, 0.0)(1.0, 0.41)(0.71, 0.71) \qbezier(0.71, 0.71)(0.41, 1.0)(0.0, 1.0) \end{picture} $

Will you please explain how did you get that picture(how it is linked to the question)?

**malaygoel** · June 20th 2006, 06:53 PM

Originally Posted by malaygoel

Will you please explain how did you get that picture(how it is linked to the question)?

Ok, I understand it.
I drew the graphs of
$x + y = 1$
$x^2 + y^2 = 1$
and got the same picture as yours.
I got the answer also from the picture.
Did you also arrive at the picture using Analytic Geometry?If no, what method you adopted.

Thanks for your help,JakeD

KeepSmiling
Malay

**JakeD** · June 20th 2006, 08:09 PM

Originally Posted by malaygoel

Ok, I understand it.
I drew the graphs of
$x + y = 1$
$x^2 + y^2 = 1$
and got the same picture as yours.
I got the answer also from the picture.
Did you also arrive at the picture using Analytic Geometry?If no, what method you adopted.

Thanks for your help,JakeD

KeepSmiling
Malay

I used the same thinking. Glad to help. --JakeD

**malaygoel** · June 21st 2006, 02:32 AM

Originally Posted by JakeD

I used the same thinking. Glad to help. --JakeD

But the problem is that we are discouraged to use Analytic Geometry.
Could the answer be obtained without using Analytic Geometry?

Keep Smiling
Malay

**ThePerfectHacker** · June 21st 2006, 05:46 AM

Originally Posted by CaptainBlack

Have you considered revising you signature to :

1) do not make posts of cardinality greater than 1
2) do not discuss illegal operations on the forum
3) do not use improper integrals
4) do not abuse notation
5) Axiom of Choice is allowed.

RonL

I like that

Maybe once we have rules on the forum link, then I will take it off.

**JakeD** · June 21st 2006, 07:05 AM

Originally Posted by malaygoel

But the problem is that we are discouraged to use Analytic Geometry.
Could the answer be obtained without using Analytic Geometry?

Keep Smiling
Malay

Yes. Use integral calculus to find the area of the region

$\begin{aligned} x + y &\ge 1 \\ x^2 + y^2 &\le 1 \\ 0 \le x &\le 1 \\ 0 \le y &\le 1. \\ \end{aligned}$

**malaygoel** · June 21st 2006, 08:42 AM

Originally Posted by JakeD

Yes. Use integral calculus to find the area of the region

$\begin{aligned} x + y &\ge 1 \\ x^2 + y^2 &\le 1 \\ 0 \le x &\le 1 \\ 0 \le y &\le 1. \\ \end{aligned}$

Thanks, but what I wanted to say that can the answer be obtained without using analytic geometry at any stage?
We have used analytic geometry to obtain the picture!

Keep Smiling

Malay

**JakeD** · June 21st 2006, 09:13 AM

Originally Posted by malaygoel

Thanks, but what I wanted to say that can the answer be obtained without using analytic geometry at any stage?
We have used analytic geometry to obtain the picture!

Keep Smiling

Malay

The inequalities involving $x$ and $y$ come from considering what it means to be an obtuse triangle. That comes from geometry. But the picture comes from graphing those inequalities. Then the leap to the probability comes from seeing that the picture is of a unit circle and a triangle. That analytical geometry and trigonometry is not necessary. As I said you can use calculus there. But then the trig comes back in solving the integral for the circle. The answer contains $\pi$ remember.

**malaygoel** · June 21st 2006, 06:03 PM

Originally Posted by JakeD

The inequalities involving $x$ and $y$ come from considering what it means to be an obtuse triangle. That comes from geometry. But the picture comes from graphing those inequalities.

We have to draw arough sketch, we can't use graphs.

Then the leap to the probability comes from seeing that the picture is of a unit circle and a triangle. That analytical geometry and trigonometry is not necessary. As I said you can use calculus there. But then the trig comes back in solving the integral for the circle. The answer contains $\pi$ remember.

Trigonometry can be used as extensively as you want. $\pi$ can arise from trigonometry.

Keep Smiling

Malay

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