# Find the measures of the sides of ∆ABC and classify the triangle by its sides.

A(3,-3), B(1,4), C(-1,-1)

## Well, well, well... let's measure these sides, shall we?

Using the distance formula, we can calculate the distances between the points:

AB = √[(1 - 3)^2 + (4 - (-3))^2]
BC = √[(-1 - 1)^2 + (-1 - 4)^2]
CA = √[(3 - (-1))^2 + (-3 - (-1))^2]

After calculating these distances, we find that AB ≈ 7.28, BC ≈ 5.83, and CA ≈ 7.21 (rounded to two decimal places).

Now, let's examine the triangle by its sides.

Since all three sides have different lengths, we have a special name for this type of triangle: a scalene triangle. It's the fancy way of saying that none of the sides are the same.

So, ∆ABC is a scalene triangle! Happy measuring!

## To find the measures of the sides of ∆ABC, we can use the distance formula. The distance formula is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the lengths of the sides of ∆ABC:

Side AB:
d(AB) = √((1 - 3)^2 + (4 - (-3))^2)
= √((-2)^2 + (7)^2)
= √(4 + 49)
= √53

Side BC:
d(BC) = √((-1 - 1)^2 + (-1 - 4)^2)
= √((-2)^2 + (-5)^2)
= √(4 + 25)
= √29

Side AC:
d(AC) = √((-1 - 3)^2 + (-1 - (-3))^2)
= √((-4)^2 + (2)^2)
= √(16 + 4)
= √20
= 2√5

Now, let's classify the triangle by its sides:

Since all three sides have different lengths, ∆ABC is a scalene triangle.

## To find the measures of the sides of triangle ABC, we need to use the distance formula. The distance formula gives us the distance between two points in a coordinate plane.

The distance formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

Let's calculate the length of each side of triangle ABC using the distance formula:

Side AB:
dAB = √((x2 - x1)² + (y2 - y1)²)
= √((1 - 3)² + (4 - (-3))²)
= √((-2)² + (7)²)
= √(4 + 49)
= √53

Side BC:
dBC = √((x2 - x1)² + (y2 - y1)²)
= √((-1 - 1)² + (-1 - 4)²)
= √((-2)² + (-5)²)
= √(4 + 25)
= √29

Side AC:
dAC = √((x2 - x1)² + (y2 - y1)²)
= √((-1 - 3)² + (-1 - (-3))²)
= √((-4)² + (2)²)
= √(16 + 4)
= √20
= √(4 * 5)
= 2√5

Therefore, the measures of the sides of triangle ABC are:
AB = √53
BC = √29
AC = 2√5

To classify the triangle by its sides, we can compare the lengths of the sides.

If all three sides are equal, the triangle is equilateral.
If only two sides are equal, the triangle is isosceles.
If all three sides are different lengths, the triangle is scalene.

Based on the lengths we calculated, since AB, BC, and AC have different lengths, triangle ABC is a scalene triangle.

## use the distance formula for each side. For example,

AB = √((1-3)^2+(4+3)^2) = √53
and so on.

Now, if the sides in order of length are a,b,c then if

c^2 = a^2+b^2 then you have a right triangle
if less, then acute
if more, then obtuse.

If all sides are different, then it is scalene

If two sides are equal, the it is isosceles

If three sides are equal, then it is equilateral