What is true about △abc? check all that apply.

1)ab ⊥ ac
2)the triangle is a right triangle.
3)the triangle is an isosceles triangle.
4)the triangle is an equilateral triangle.
5)bc ∥ ac

Answers and explanation
your answers are:
1) ab ⊥ ac
2) the triangle is a right triangle. 
3) the triangle is an isosceles triangle. 


the distance between points a and b (length of segment ab) can be found using the distance formula (really an application of the pythagorean theorem):

   

the distance between points a and c (length of segment ac) can be found using the distance formula

   

the distance between points b and c (length of segment bc)  can be found by using the distance formula (with the y-coordinate part omitted)

   

to show that this is a right-triangle, we can show that the pythagorean theorem holds for all the triangle lengths:

   

this equation holds true, therefore, " 2)the triangle is a right triangle". bc is the hypotenuse.

" 3)the triangle is an isosceles triangle"  because the length of ab is equal to the length of ac, and they're not equal to the length of bc.

since this is a right triangle,  ∠bac must be a right angle, thus making " 1) ab ⊥ ac" true . this can also be proved using slopes. we use the fact that slope = rise/run.

for ab, the "rise" is +1 and the run is +1 to get to the next "nice" point, therefore the slope of segment ab is +1.

for ac, we have the "rise" +1 but run -1 (in negative direction) to get to the next "nice" point. therefore, the slope of ac is -1.

slopes of ab and ac are negative reciprocals, so ab is perpendicular to ac.

for the other choices:
this is not an equilateral triangle because all the side lengths are not equal.
bc and ac are not parallel lines.

answer:

step-by-step explanation:

from the given figure, ab= (using distance formula)

ab=

also, ac=

ac=

the triangle is an isosceles triangle because the length of ab is equal to the length of ac, and they're not equal to the length of bc.

now, bc=

bc=8

using the pythagoras theorem,

thus,the equation holds. the triangle is a right triangle. bc is the hypotenuse.

since this is a right triangle, ∠bac must be a right angle,thus, ab ⊥ ac.

this is not an equilateral triangle because all the side lengths are not equal.

also, bc and ac are not parallel lines.

hence, option (1), (2) aqnd (3) is correct.