Show that the points form the vertices of the polygon.

Question

isosceles triangle:    (2, 2), (4, 8), (−2, 6)

distance between (2, 2) and (4, 8) d1=__________units?

distance between (4, 8) and (−2, 6) d2=_________units?

distance between (2, 2) and (−2, 6) d3=_________units?

Answer 1

There are three points given. They are:

A(2, 2)

B(4, 8)

C(−2, 6)

It usually helps a lot to plot these as (x,y) on graph paper. Even a rough sketch is helpful.

For the three line segments connecting these points (that is, AB, BC, and CA), a triangle may be formed or not. The condition is: the sum of any two lengths must exceed the third length.

So, we need the distance formula:

d_AB = √ ( (y_B - y_A)² + (x_B - x_A)² ) [notice that this is the Pythagorean Theorem c² = a² + b² ]

d₁ = length of AB = √ ( (8-2)² + (4-2)² ) = √ (6² + 2²) = √ (36+4) = √40 = 2√10

d₂ = length of BC = √ ( (6-8)² + ((-2)-4)² ) = √ ( (-2)² + (-6)² ) = √(4+36) = √40 = 2√10

d₃ = length of CA = √ ( (2-6)² + (2-(-2))² ) = √ ( (-4)² + 4² ) = √(16+16) = √32 = 4√2

The definition of isosceles triangle : two sides of equal length

ABC is an isosceles triangle.

Note:

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.