Question

D(5,7) E (4,3) and f(8,2) from the vertices of a triangle what is m DEF

Answer 1

m<DEF = 90°

Explanation:

It is given that, D, E and F are the vertices of a triangle.

D(5,7) E (4,3) and f(8,2)

To find side DE

D(5,7) E (4,3)

DE = √[(5-4)² + (7 - 3)²] = √17

To find side EF

E (4,3) ,F(8,2)

EF = √[(8-4)² + (2 - 3)²] = √17

To find side DF

D(5,7) ,F(8,2)

EF = √[(8-5)² + (2 - 7)²] = √34 = √2√17

To find the ratio of sides

The sides are in the ratio

DE : EF : DF = √17 : √17 : √2√17 = 1 : 1 : √2

To find the angle DEF

The sides are in the ratio 1 : 1 : √2

Therefore triangle DEF is a right angled triangle.

DE = EF

<DEF = 90°

Answer 2

m∠DEF = 90°

Explanation:

As we know to determine the angle between two lines with the given equations or vertices we use the formula

`tan\theta =(m_(2)-m_(1))/(1+m_(2)m_(1))`

Now we have been given the vertices of a triangle as D(5, 7) E(4, 3) F(8, 2)

To measure m∠ DEF will use the formula

Since slope of DE
`m_(1)=(y_(2)-y_(1))/(x_(2)-x_(1))`

`m_(1)=(7-3)/(5-4)=(4)/(1)=4`

Slope of EF
`m_(2)=(3-2)/(4-8)=(1)/(-4)=-(1)/(4)`

Now m∠DEF =
`tan\theta = ((17)/(4))/(1-1)=\infty`

Therefore
`\theta =90`

Answer is ∠DEF = 90°