Given: A(8, 9), B(10, 3) and C(3, 4). a. Prove ABC is isosceles. b. Find M, the midpoint of AB. c. Prove CM AB.

Question

Given: A(8, 9), B(10, 3) and C(3, 4).

a. Prove ABC is isosceles.

b. Find M, the midpoint of AB.

c. Prove CM perp. to AB.

Answer 1

A) Prove this triangle is isosceles.

OK, the first thing to do is recall the definition of an isosceles triangle: this is a triangle in which two legs have the same length. So to prove this, you must find the lengths of all 3 sides of the triangle.

The length of each side is the same is the distance between the vertices, so here you will use the distance formula:

distance = √[(x₁ - x₂)² + (y₁ - y₂)²]

In words, this is, add (the difference in x coordinates, squared )

plus (the difference in x coordinates, squared )

and then take the square root of that sum.

So for side AB, this is: √[(8-10)² + (9-3)²]

= √[(-2)² + (6)²]

= √[4 + 36]

= √40

Then you must continue this process for sides BC and AC, and see what you find!

B) The midpoint of a line segment will have an x-value that is in the middle of the endpoints' x-values. Likewise, its y-value is in the middle of the endpoints' y-values. Sometimes people say, the x is the average of the endpoints x-values, and the y is the average of the endpoints' y-values. Same thing.

It is calculated with the formula:

midpoint(x) = [end1(x) + end2(x)] ÷ 2

Here, M's x value will be: [8 + 10] ÷ 2

= [18] ÷ 2

= 9

And you can do the same process for M's y-value.

C) Here you will need to compare the slopes of CM and AB.

Slope is "rise over run", or the change in y-values divided by the change in x-values.

*It's really important here that you decide which endpoint you will use first, and which second, and do this the same way for the x-values and the y-values.*

The rise is: end1(y) - end2(y)

The run is: end1(x) - end2(x)

So in this example, the slope of AB can be calculated as:

rise = 9 - 3 = 6

run = 8 - 10 = -2

slope = rise ÷ run = 6 ÷ -2 = -3

To determine if lines (or segments) are perpendicular, you must use their slopes. Perpendicular lines' slopes are *opposite reciprocals*. That means, if one slope is a / b, the other is -b / a. One slope is positive, and the other must be negative. And the numbers of the slopes are, as fractions, one is the upside-down version of the other. An example with numbers: if one slope is -1/2, the other is 2/1 ( = 2).

Now, you must show your work to prove that the slope of CM is the opposite reciprocal of -3, which is 1/3.