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Question

Prove that the points (0,0),(5,5) and (-5,5) are the vertices of a right isosceles triangle.


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Solution

STEP 1 : Assumption

Let A(0,0),B(5,5) and C(-5,5) be the given points.

STEP 2 : Finding the distance between the three points

By distance formula distance between two points P(x1,y1) and Q(x2,y2) is

PQ=x2-x12+y2-y12

Therefore distance between AB,BC and CA can be found using the above formula

AB=(5-0)2+(5-0)2

AB=25+25

AB=50

BC=(-5-5)2+(5-5)2

BC=102+0

BC=100

CA=0--52+(0-5)2

CA=0+52+(0-5)2

CA=52+(-5)2

CA=25+25

CA=50

Since, AB=AC

Hence, ABC is an isosceles triangle.

STEP 3 : Finding the relation between AB,BC and CA

AB2=50

BC2=100

CA2=50

BC2=AB2+CA2

Hence, ABC is a right angles triangle.

Since, AB=AC and BC2=AB2+CA2
Therefore, ABC is a right isosceles triangle.


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