Question

The vertices of a triangle are J(1,1) K(2,4) L(5,3) . Find the slopes of the lines connecting the vertices. Then determine whether the triangle is a right triangle.

Answer 1

Yes, the triangle is a right triangle.

Step-by-step explanation:

Since we know that slope formula of the line connecting two points is:
`m=(y_2-y_1)/(x_2-x_1)`, where,

m = Slope of the line.

`y_2-y_1` = Difference between y-coordinates of two points.

`x_2-x_1` = Difference between corresponding x-coordinates of same two points.

Let us find slope of the line connecting points J(1,1) and K(2,4).

`\text{Slope of the line connecting points J and K}=(4-1)/(2-1)`

`\text{Slope of the line connecting points J and K}=(3)/(1)=3`

Let us find slope of the line connecting points K(2,4) and L(5,3).

`\text{Slope of the line connecting points K and L}=(3-4)/(5-2)`

`\text{Slope of the line connecting points K and L}=(-1)/(3)`

Let us find slope of the line connecting points L(5,3) and J(1,1).

`\text{Slope of the line connecting points L and J}=(1-3)/(1-5)`

`\text{Slope of the line connecting points L and J}=(-2)/(-4)`

`\text{Slope of the line connecting points L and J}=(1)/(2)`

Since we know that perpendicular lines intersect at right angles to one another and the slopes of perpendicular lines are opposite reciprocals of each other.

We can see that slope of line connecting points J and K is 3, while slope of the line connecting points K and L is
`-(1)/(3)`.

Since
`-(1)/(3)` is negative reciprocal of 3, therefore, the triangle is a right triangle.

Answer 2

The triangle is not a right triangle.

Explanation:

The given triangle has vertices
`J(1,1),K(2,4),L(5,3).`

The formula for finding the slope is

`m=(y_2-y_1)/(x_2-x_1)`

The slope of JK is

`m_(JK)=(4-1)/(2-1)=3`

The slope of JL is

`m_(JL)=(5-1)/(3-1)=2`

The slope of KL is

`m_(KL)=(5-4)/(3-2)=1`

The products of the slopes of the adjacent sides are

`m_(KL)* m_(JL)=1* 2=2`

`m_(KL)* m_(JK)=1* 3=3`

`m_(JK)* m_(JL)=3* 2=6`

None of the products is
`-1`, which means that no adjacent sides are perpendicular. Hence the triangle is not a right triangle.