Question

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

Answer 1

Answer:

m∠DEF = 90°

Step-by-step explanation:

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

$tan\theta =\frac{m_{2}-m_{1}}{1+m_{2}m_{1}}$

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 $m_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m_{1}=\frac{7-3}{5-4}=\frac{4}{1}=4$

Slope of EF $m_{2}=\frac{3-2}{4-8}=\frac{1}{-4}=-\frac{1}{4}$

Now m∠DEF = $tan\theta = \frac{\frac{17}{4}}{1-1}=\infty$

Therefore $\theta =90$

Answer is ∠DEF = 90°

Answer 2

Answer:

m<DEF = 90°

Step-by-step 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°