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Question

Find a relationship between $$x$$ and $$y$$ so that the triangle whose vertices are given by $$(x,y),(1,1)$$ and $$(5,1)$$ is a right triangle with the hypotenuse defined by the points $$(1,1)$$ and $$(5,1)$$.


A
(x+3)2(y1)2=22
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B
(x3)2+(y1)2=22
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C
x2+y2=22
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D
(x3)2(y2)2=32
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Solution

The correct option is B $$(x-3)^2+(y-1)^2=2^2$$
Let us use the distance formula to find the length of the hypotenuse $$h$$ with points $$(1,1)$$ and $$(5,1)$$. 
$$h=\sqrt { \left( 5-1 \right) ^{ 2 }+\left( 1-1 \right) ^{ 2 } } =\sqrt { (4)^{ 2 }+\left( 0 \right) ^{ 2 } } =\sqrt { 16 } =4$$ 
We now use the distance formula to find the sizes of the two other sides $$a$$ and $$b$$ of the triangle. 
$$a=\sqrt { \left( x-1 \right) ^{ 2 }+\left( y-1 \right) ^{ 2 } } $$ 
$$b=\sqrt { \left( x-5 \right) ^{ 2 }+\left( y-1 \right) ^{ 2 } } $$ 
By pythagoras theorem gives 
$$\left( 4 \right) ^{ 2 }=\left( \sqrt { (x-1)^{ 2 }+\left( y-1 \right) ^{ 2 } }  \right) ^{ 2 }+\left( \sqrt { \left( x-5 \right) ^{ 2 }+\left( y-1 \right) ^{ 2 } }  \right) ^{ 2 }$$
Expand the squares, simplify and complete the squares to rewrite the above relationship between $$x$$ and $$y$$ as follows. 
$$(x - 3)^ 2 + (y - 1)^ 2 = 2^ 2$$

Mathematics

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