Question

The hypotenuse of a right triangle has endpoints A(4, 1) and B(–1, –2). On a coordinate plane, line A B has points (4, 1) and (negative 1, negative 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:

Answer 2

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

Explanation:

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

To find: coordinates of vertex of the right angle

Solution:

Let C be point
`(x,y)`

Distance between points
`(x_1,y_1),(x_2,y_2)` is given by
`√((x_2-x_1)^2+(y_2-y_1)^2)`

`AC=√((x-4)^2+(y-1)^2)\\BC=√((x+1)^2+(y+2)^2)\\AB=√((4+1)^2+(1+2)^2)=√(25+9)=√(34)`

ΔABC is a right angled triangle, suing Pythagoras theorem (square of hypotenuse is equal to sum of squares of base and perpendicular)

`34=\left [ (x-4)^2+(y-1)^2 \right ]+\left [ (x+1)^2+(y+2)^2 \right ]`

Put
`(x,y)=(-1,1)`

`34=\left [ (-1-4)^2+(1-1)^2 \right ]+\left [ (-1+1)^2+(1+2)^2 \right ]\\34=25+9\\34=34`

which is true. So,
`(-1,1)` can be a vertex

Put
`(x,y)=(4,-2)`

`34=\left [ (4-4)^2+(-2-1)^2 \right ]+\left [ (4+1)^2+(-2+2)^2 \right ]\\34=9+25\\34=34`

which is true. So,
`(4,-2)` can be a vertex

Put
`(x,y)=(1,1)`

`34=\left [ (1-4)^2+(1-1)^2 \right ]+\left [ (1+1)^2+(1+2)^2 \right ]\\34=9+4+9\\34=22`

which is not true. So,
`(1,1)` cannot be a vertex

Put
`(x,y)=(2,-2)`

`34=\left [ (2-4)^2+(-2-1)^2 \right ]+\left [ (2+1)^2+(-2+2)^2 \right ]\\34=4+9+9\\34=22`

which is not true. So,
`(2,-2)` cannot be a vertex

Put
`(x,y)=(4,-1)`

`34=\left [ (4-4)^2+(-1-1)^2 \right ]+\left [ (4+1)^2+(-1+2)^2 \right ]\\34=4+25+1\\34=30`

which is not true. So,
`(4,-1)` cannot be a vertex

Put
`(x,y)=(-1,4)`

`34=\left [ (-1-4)^2+(4-1)^2 \right ]+\left [ (-1+1)^2+(4+2)^2 \right ]\\34=25+9+36\\34=70`

which is not true. So,
`(-1,4)` cannot be a vertex

So, possible points for the vertex are
`(-1,1),(4,-2)`