Question

What type of triangle is formed by joining the points D(7, 3), E(8, 1), and F(4,-1)?

Answer 1

A right-angled triangle is formed.

Explanation:

1) calculate d the distance between the points of the triangles, use the distance formula :
`d=sqrt((x2−x1)^2+(y2−y1)^2`

`D(7,3), E(8,1),F(4,-1)`

`=> DE=sqrt((8-7)^2+(1-3)^2))=sqrt(1^2+2^2)=sqrt5`
`=> EF=sqrt((4-8)^2+(-1-1)^2)=sqrt(4^2+2^2)=sqrt20`
`=>DF==sqrt((4-7)^2+(-1-3)^2)=sqrt(3^2+4^2)=sqrt25=5`

Pythagorean theorem : `c^2=a^2+b^2`

The theorem states that: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.

Now we use the theorem to check and see if the triangle is right-angled or not :

`=> sqrt(25)^2=sqrt5^2+sqrt20^2`
`=> 25=5+20`

As `(DF)^2=(DE)^2+(EF)^2`, triangle `DEF` is right-angled.