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Re: Point (x,y) is a point within the triangle. [#permalink]

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23 Sep 2012, 18:19

In the given triangle y<x, when value of x => 3.35(approx.)

So, we we form a triangle by taking this point(3.35,3.3) in the given triangle. The solution to above question will be the area of the smaller triangle divided by the larger one.

Area of smaller triangle(2.85)/Area of larger triangle(25)

Answer is B (1/8).

I am not sure of my approach. So experts, please can you advise if my approach is correct.

The small triangle would be made up of 3 points: 1. the origin 2. (5,0), 3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = \(\frac{10}{3}\) = y

The small triangle would be made up of 3 points: 1. the origin 2. (5,0), 3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = \(\frac{10}{3}\) = y

The small triangle would be made up of 3 points: 1. the origin 2. (5,0), 3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = \(\frac{10}{3}\) = y

Hi guys.. Can you please explain how did you find the equation of hypotenuse.. y=mx+c >> m=-2 , why c= 10 ?

When m and c are not 0, the line is not horizontal and will not pass through the origin. Then both the x and the y intercept will be non-zero. The y intercept is the value of y for x = 0, which, for the equation y = mx + c, is c. The x intercept is the value of x for y = 0, which is -c/m. The given equation y = mx + c can be rewritten as -mx + y = c, or \(\frac{x}{-c/m}+\frac{y}{c}=1\). You can see that the denominator of x is exactly the x intercept and the denominator of y is the y intercept.

Each line which doesn't go through the origin, has its equation as \(\frac{x}{x_i}+\frac{y}{y_i}=1\) , where \(x_i\) and \(y_i\) are the x and the y intercept, respectively.

In our case, we could have written directly the equation of the hypotenuse as \(\frac{x}{5}+\frac{y}{10}=1\) which we can rearrange and get \(y=-2x+10.\)

So, next time, if you have the two intercepts, for example you know that the line goes through the points (-3,0) and (0,4), you can immediately write the equation of the line as \(\frac{x}{-3}+\frac{y}{4}=1\) rearrange as you wish... I mean you can save the time of finding the slope and write the standard equation of a line... Not that it is such a saving, but anyway, it is a nice mathematical property
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Re: Point (x,y) is a point within the triangle. [#permalink]

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05 Jan 2013, 02:43

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saikarthikreddy wrote:

Point (x,y) is a point within the triangle. What is the probability that y<x?

a. 1/4 b. 1/8 c. 1/6 d. 1/2 e. 1/5

1. Use the x=y boundary line. The region of the triangle below this line contains points x > y. 2. Get the line that of the triangle.

\(m = \frac{10 - 0}{0-5} = -2\)

\(y = -2x + b\) \(10 = -2(0) + b\) \(b=10\)

\(Line: y = -2x + 10\)

3. Get the point of intersection of y=x and y=-2x+10. \(x = -2x + 10\) \(3x = 10\) \(x = 10/3\)

4. Get the area of smaller triangle: \(=\frac{10}{3}*\frac{1}{2}*5=\frac{25}{3}\) 5. Get the area of the larger triangle: \(10*5*\frac{1}{2} = 25\) 6. \(\frac{smallerArea}{largerArea}=\frac{25}{3}*\frac{1}{25}=\frac{1}{3}\)

Re: Point (x,y) is a point within the triangle. [#permalink]

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07 Jan 2013, 17:21

Shrek89 wrote:

Why are u guys calculating the triangle area.. with 1/2 *b*h ???

Its not a right triangle right??

We have to get the portion of the triangle (0,0), (10,0) and (5,0) with x>y. A boundary line x=y will divide this triangle to two portions: a portion with x>y and a portion with x<y. Now you have to get the desired portion which is the smaller triangle below the x=y boundary but still within the main triangle. That's why we are calculating two areas: \(\frac{desired portion}{main triangle}\)
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Re: Point (x,y) is a point within the triangle. [#permalink]

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07 Jan 2013, 19:53

That is a good question. But you can still calculate even if it is not a right triangle.

The smaller triangle is formed by the coordinates (0,0), (10/3,10/3) and (5,0). This is not a right triangle but you are given its height through its (10/3,10/3) coordinate. Then your base is 5 which is equal to the base of your main triangle.

The small triangle would be made up of 3 points: 1. the origin 2. (5,0), 3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = \(\frac{10}{3}\) = y

Hi, What if the question is "what is the probability that y>x?" Then, is the answer 2/3 correct? One way to calculate is 1-1/3. But, if the question directly asks the prob. of y>x, by using your method I got 2/3. I am sure this is right, but please confirm.

The small triangle would be made up of 3 points: 1. the origin 2. (5,0), 3. a point on the hypotenuse where y=x

To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = \(\frac{10}{3}\) = y

Hi, What if the question is "what is the probability that y>x?" Then, is the answer 2/3 correct? One way to calculate is 1-1/3. But, if the question directly asks the prob. of y>x, by using your method I got 2/3. I am sure this is right, but please confirm.

Re: Point (x,y) is a point within the triangle. What is the [#permalink]

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21 Oct 2014, 19:32

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so one way to think of this is the area of the triangle where x>y over the total area of the triangle.

Let's start with the total area, which is 25

The ares of the triangle can be discovered by first finding where x=y and the hypotenuse meet. We know two points on the hypotenuse (0,10) and (5,0), so we know the y intercept is 10 and the slope is (-10/5) or -2. y=-2x+10

That meets y=x when x=-2x+10 or when 3x=10, so x=10/3 and y =10/3

we now know the height of the triangle (10/3) and the base of the triangle 5, so the total area is 50/6 or 25/3

That divided by 25 (the total area of the triangle) equals 1/3 (B)
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Re: Point (x,y) is a point within the triangle. What is the [#permalink]

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05 Dec 2015, 11:19

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