Is the triangle a right triangle? Explain your reasoning. (use description to answer)

Question

A (3, 5), B (-2, 6), and C (0, 16) are the vertices of a triangle.

Answer 1

Method1: Length method

First we need to calculate the length of each side:

AB = [(-2-3)^2 + (6-5)^2 ]^0.5 = (25+1)^0.5 = 26^0.5

BC = [(-2-0)^2 + (6-16)^2 ]^0.5 = (4+100)^0.5 =104^0.5

AC = [(0-3)^2 + (16-5)^2 ]^0.5 = (9+121)^0.5 = 130^0.5

Line AC is the longest, and assume it is a hypotenuse : First check if it a triangle

AC > AB + BC => 130^0.5 > 26^0.5 + 104^0.5 => it is a triangle, the we go ahead and check if

it is a right triangle

AC^2 = AB^2 + BC^2

=> (130^0.5)^2 + (104^0.5)^2 = (26^0.5)^2

=> 130 = 104 + 26 = 130 => It is a right triangle AC is hypotenuse

Method2 Slope method:

Calculate the slope of each line

Slope of line AB = (6-5)/(-2-5) = -1/5

Slope of line BC = (16-6)/(0+2) = 5

Slope of line AC = (16-5)/(0-3) = -5

(slope of AB)x(slope of BC) = -1/5 x 5 = -1

Hence angle formed by lines AB and BC is right angle

Answer 2

Using distance formula

AB = Sqrt ( 26)

BC = Sqrt ( 104)

AC = Sqrt (130)

Using Pythagorean Theorem

Angle ABC is rt angle

as 26 + 104 = 130

Answer 3

Draw the triangle.

Determine which sides might be perpendicular and form a right angle.

Determine the slopes of those two sides using the coordinates of the endpoints.

If the product of the two slopes is negative, then the lines are perpendicular and form a right angle, and the triangle is a right triangle.