Question

(0,−2),(−5,4) and (5,4) Determine whether or not the given points form a right triangle. If the triangle is not a right triangle, determine if it is isosceles or scalene

Answer 1

Isosceles

Explanation:

Use distance formula:

d =
`\sqrt{(x_(2) - x_(1))^2 + (y_(2) - y_(1))^2 }`

Find distance between each point, there should be 3 lengths given after we find it. Once we find the length of each side, we will use the Pythagorean theorem to check and see if it is a right angle.

Let's first find out what each of these terms mean:

Right triangle: A triangle with one angle measuring 90 degrees.

Scalene triangle: A triangle with all sides not equal to each other at all.

Isosceles triangle: A triangle that has two sides of equal length.

Now let's find the distance between all sets of pairs of coordinates when (0, -2) to (-5, 4) form a length and (-5, 4) to (5, 4) form another length, and finally (5, 4) to (0, -2) form the last length.

We could also verify this with a graph in the end.

d =
`\sqrt{(x_(2) - x_(1))^2 + (y_(2) - y_(1))^2 }`

(0, -2) to (-5, 4) are our first coordinate set to solve, we'll call (0, -2) the first pair and (-5, 4) the second:

Substitute:

d =
`√((-5 - 0)^2 + (4 - (-2))^2 )`

Evaluate parenthesis first:

d =
`√((-5)^2 + (4 + 2)^2 )`

d =
`√((-5)^2 + (6)^2 )`

Solve exponents:

d =
`√(25 + 36 )`

d =
`√(61)`

Let's keep
`√(61)` under a radical to compare lengths in the end.

Now to find our second length, we'll call (-5, 4) our first pair and (5, 4) our second:

d =
`\sqrt{(x_(2) - x_(1))^2 + (y_(2) - y_(1))^2 }`

d =
`√((5 - (-5))^2 + (4 - 4)^2 )`

d =
`√((5 + 5)^2 + (0)^2 )`

d =
`√((10)^2 + (0)^2 )`

d =
`√(100 + 0)`

d =
`√(100)`

Which square roots perfectly into:

d = 10

Finally, the third length,

we'll call (5, 4) our first pair and (0, -2) our second:

d =
`\sqrt{(x_(2) - x_(1))^2 + (y_(2) - y_(1))^2 }`

d =
`√((0 - 5)^2 + (-2 - 4)^2 )`

d =
`√((-5)^2 + (-6)^2 )`

d =
`√(25 + 36)`

d =
`√(61)`

Now let's compare all the lengths together:

`√(61)`,
`10`,
`√(61)`

We can see that the first and third lengths are the same(
`√(61)` and
`√(61)`). And judging from all the classifications of a triangle, this seems to fit the category of an isosceles triangle!

Two lengths of a triangle that are congruent(the same) are an isosceles triangle.

Therefore this triangle is isosceles.