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. $(x+3{)}^2-(y-1{)}^2=2^2$
• B. $(x-3{)}^2+(y-1{)}^2=2^2$
• C. $x^2+y^2=2^2$
• D. $(x-3{)}^2-(y-2{)}^2=3^2$

Answer 1

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 $(5,1)$.

$h=\sqrt{{(5-1)}^2+{(1-1)}^2}=\sqrt{(4{)}^2+{\left(0\right)}^2}=\sqrt{16}=4$

We now use the distance formula to find the sizes of the two other sides $a$ and $b$ of the triangle.

$a=\sqrt{{(x-1)}^2+{(y-1)}^2}$

$b=\sqrt{{(x-5)}^2+{(y-1)}^2}$

By pythagoras theorem gives

${\left(4\right)}^2={\left(\sqrt{(x-1{)}^2+{(y-1)}^2}\right)}^2+{\left(\sqrt{{(x-5)}^2+{(y-1)}^2}\right)}^2$

Expand the squares, simplify and complete the squares to rewrite the above relationship between $x$ and $y$ as follows.

$(x-3{)}^2+(y-1{)}^2=2^2$