Using the Distance Formula

Question

Find the distance between (3a + 7, b) and (b + 7, 3a), given than a > b

Answer 1

Ok, so think of this as finding the hypotenuse C of a right triangle, with side A as (x-x₁) <---distance/length between the X coordinates; and side B as (y-y₁)<---distance/length between the Y coordinates. If you connect the 2 points, it forms the hypotenuse, or slanted side (across from the right angle), of a right triangle. You can draw the other sides by drawing a line straight down from the "top point" and straight across from "bottom point" until they connect to finish the triangle.

The Pythagorean Theorem helps us find a missing side length of a right triangle, given the lengths of the other 2 sides. Reminder, here is the Pythagorean Theorem: a²+b²=c² (again, C is the hypotenuse, or slanted side that is across from the right angle in the triangle). Through algebra, we can rewrite this formula as: C=square root of (a²+b²); Given a is side length of the X coordinates and b is the side length of the Y coordinates, you can again, rewrite it to fit the situation like so: C=square root of [(x-x₁)² + (y-y₁)²)] In this case, here would be the equation using the given information from the original problem:

C= square root of {[(3a + 7)-(b + 7)]²+(b-3a)²}

You will then solve from there. The best way to do this is to simplify it as much as possible first. Let's start with the first brackets:

(3a+7)-(b+7)---->3a+7-b-7---->3a-b so, inside the first set of brackets, we'll have 3a-b

The 2nd set of brackets is as simple as it gets, so let's rewrite the equation with the simplified bracket in place:

C= square root of {(3a-b)²+(b-3a)²} This should be much easier to look at!

Next, let's square the first set of ( ) (representing x-x₁, or the A side)

(3a-b)²---->(3a-b)(3a-b)---->9a²-3ab-3ab+b²---->9a²-6ab+b²

Now, let's square the 2nd set of ( ) (representing y-y₁, or the B side)

(b-3a)²---->(b-3a)(b-3a)---->b²-3ab-3ab+9a²---->b²-6ab+9a²

Remember, we will need to add these 2 expressions together, based on the underlined equation above, so let's do that next. We will basically be combining "like" terms here:

(9a²-6ab+b²)+(b²-6ab+9a²)---->9a²+9a²-6ab-6ab+b²⁺b²---->18a²-12ab+2b²---->2(9a²-6ab+b²)

If we put this back into the equation, we have the answer of:

C=square root of [2(9a²-6ab+b²)] <----This is the distance between the 2 points given in the original problem