Triangle def has vertices d(2,-5) e(6,-1) and f(5,6) verify that def is isosceles
and write an equation of the bisector of angle f

To prove that the given triangle is an isosceles, two of its sides must have equal lengths.

Between D(2,-5) and E(6,-1):
d = √(6 - 2)² + (-1 - ⁻5)² = 4√2

Between E(6,-1) and F(5,6):
d = √(5 - 6)² + (6 - ⁻1)² = 5√2

Between D(2,-5) and F(5,6)
d = √(5 - 2)² + (6 - ⁻5)² = √130

Since there are no equal lengths, the triangle is NOT an isosceles triangle.

The bisector of angle F is a line segment from vertex F to the midpoint of line DE.

Midpoint: x = (2+6)/2 = 4; y = (-5+-1)/2 = -3
The midpoint is at (4,-3).
The equation of the line follows the point-slope formula:
y - y₁ = [(y₂ - y₁)/(x₂ - x₁)](x - x₁)
Substituting using points (5,6) and (4,-3)
y - 6 = [(⁻3 - 6)/(4 - 6)](x - 5)
y - 6 = (9/2)(x - 5)
2(y - 6) = 9(x - 5)
2y - 12 = 9x - 45
2y - 9x = -33 

Okay so for every triangle has three angles that always add up to 180 degrees. and to tell the difference between whether they are secure, right or obtuse is that acute: has one angle that is less than 90 degrees. right triangle means there is at least one angle that is exactly 90 degrees. and an obtuse angle has an angle that is bigger than 180 degrees. i’ll do #13 as an example: 13. 40 and 50 so if every triangle has 180 degrees 40+50= 90 degrees then180-90= 90 degrees so the third angle is 90 degrees and because this triangle has a 90 degree angle it means it is a right triangle