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# prove that we get a right angled triangle by joining the points (2,1)(3,4)(-3,6)?

Posted in Math, asked by ishan, 6 years ago. 2596 hits.

## 4

There are probably a lot of ways to solve this but here is the first method I thought of:

A quick sketch suggests the point (3,4) makes an angle of 90 degrees, so we want to prove that this is the case.

First label the angles at (3,4) A, (2,1) B and (-3,6) C, then label the opposite sides to these angles a,b and c for simplicity.

The Cosine rule states that for any angle in a triangle, Cos(A)=(b2+c2-a2)/2*b*c .

So we need to find the lengths of the three sides, which can be done using Pythagoras's theorem (a2+b2=c2)and the difference between the x co-ordinates and y co-ordinates for each point.

So a = distance between (2,1) and (-3,6) =   sqrt{52 +52} = sqrt{50},

b = distance between (3,4) and (-3,6) = sqrt{62 +22} =sqrt{40},

c = distance between (3,4) and (2,1) = sqrt{12 +32} =sqrt{10}.

Now Cos(A) = (b2+c2-a2)/2*b*c = (40 + 10 - 50)/40 = 0

Finally, the only possible value for A is 90 degrees in order for Cos(A)=0, and so the statement is proved!

Hope this helps.

-used sqrt{} to stand for 'the square root of' as math editor doesnt seem to work yet

Russell Ludlow - 6 years ago
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## 1

Russell is right.

But you can also sort it out using Pythagoras Theorem.

First label the angles at (3,4) A, (2,1) B and (-3,6) C, then label the opposite sides to these angles a,b and c for simplicity.

Now find the length of each side (AB, BC && CA) using distance formula [ Distance=(sqrt(x1-x2)+ (y1-y2)2) ]

Now using pythagoras theorem : (hypotenuse)= (base)2 + (perpendicular)2

If this is verified with its sides , it is proved that it is a right angled triangle.

Gauri Narula - 6 years ago
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## 0

given coordinates,(a1,b1)    =(2,1)

(a2,b2)=(3,4)

(a3,b3)=(-3,6)

here we have one formula,area of a traingle with three coordinates=area=|a1(b2-b3)+a1(b3-b1)+a3(b1-b2)/2|

=|2(4-6)+2(6-1)+(-3)(1-4)/2|

=|2(-2)+2(5)+(-3)(-3)/2|

=|-4+10+9/2|

=|-15/2|=15/2

Venkat Nagendra Thati - 6 years ago
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- Just now

## 0

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