Question

The hypotenuse of a right triangle has endpoints A(4, 1) and B(–1, –2).

At which points could the vertex of the right angle in the triangle be located? Check all that apply.
(–1, 1)
(4, –2)
(1, 1)
(2, –2)
(4, –1)
(–1, 4)

Answer 1

(–1, 1)

(4, –2)

Explanation:

Given that the hypotenuse of a right triangle has endpoints A(4, 1) and B(–1, –2), then two triangles can be formed (see figure attached).

The vertex of the right angle in each triangle are on points (-1, 1) and (4, -2), as can be seen in the figure.

Answer 2

(–1, 1)

(4, –2)

Explanation:

Let

(x,y) ----> the coordinates of point C

Applying the Pythagoras Theorem

`(-1-4)^2+(-2-1)^2=(x-4)^2+(y-1)^2+(x+1)^2+(y+2)^2`

step 1

Find the possibles values of x

`(-1-4)^2=(x-4)^2+(x+1)^2\\\\25=x^(2)-8x+16+x^(2)+2x+1\\\\2x^(2)-6x-8=0`

Simplify

`x^(2)-3x-4=0`

Solve the quadratic equation

`x^(2)-3x-4=(x+1)(x-4)`

The values of x are

x=-1 and x=4

step 2

Find the possibles values of y

`(-2-1)^2=((y-1)^2+(y+2)^2\\\\9=y^(2)-2y+1+y^(2)+4y+4\\\\2y^(2)+2y-4=0`

Simplify

`y^(2)+y-2=0`

Solve the quadratic equation

`y^(2)+y-2=(y+2)(y-1)`

The values of y are

y=-2, y=1

therefore

The possibles coordinates of point C are

(-1,1),(4,-2)