MATH SOLVE

3 months ago

Q:
# Show that ∆ABC is a right triangle if A(0, 0), B(3, 2), and C(–2, 3). Find the slope m1 of AB and the slope m2 of AC and verify that m1m2 = –1. Provide your complete solutions and proofs.

Accepted Solution

A:

First we'll show this is a right triangle by the Pythagorean Theorem:[tex]A(0,0), B(3,2), C(-2, 3)[/tex][tex]AB^2=3^2+2^2=13[/tex][tex]AC^2=(-2)^2+3^2=13[/tex][tex]BC^2 = (-3 -3)^2 + (3 -2)^2 = 26[/tex]Since [tex]AB^2+AC^2=BC^2[/tex] we have a right triangle, by the (converse to the) Pythagorean Theorem. We also see it's isosceles, AB=AC.[tex]\textrm{Slope of AB} = m_1 = \dfrac{2 - 0}{3 - 0} = \dfrac 2 3[/tex][tex]\textrm{Slope of AC} = m_2 = \dfrac{3 - 0}{-2 - 0} = -\dfrac 3 2[/tex][tex] m_1 m_2 = (\frac 2 3)(-\frac 3 2 ) = -1 \quad\checkmark[/tex] That's the end of the homework but I'll go on a bit.Another way to show perpendicularity, essentially the same as the other two, is by a zero as the dot product of the sides as vectors, differences between vertices.Vector AB = B - A = (3,2)Vector AC = C - A = (-2, 3)AB · AC = 3(-2) + 2(3) = 0A zero dot product meansAB ⊥ AC