Here is the question again, for your .

This is the one I referred to as Problem 3.

Step-by-step explanation:

The length of OP is going to require a bit more work since we don't know P yet.

We need to find the equation for the line that travels through A & B.

Then we need to find the line that travels through O & P such that the choice in P makes OP perpendicular to AB. Perpendicular lines have opposite reciprocal slopes. The cool thing about OP is it is easy to identity the y-intercept. So the line for OP will just be . Slope-intercept form is y=mx+b where m is the slope and b is the y-intercept.

Let's go ahead and find the slope of AB.

The slope of AB can be found by using the formula:

where and are points given to you on the line.

The points given to us are: and

So entering these points into the formula gives us:

Simplifying:

Clearing the mini-fractions by multiplying top and bottom by :

.

The top is a difference of squares and so can use formula .

The bottom both terms have a common factor of so I can just factor that out of the two terms.

Let's do that:

There is a common factor to cancel:

The slope of AB is .

I'm going to use point-slope form to determine my linear equation for AB.

Point-slope form is where is slope and is a point on the line that you know.

So we have that (-1,1) is the point where is .

Plugging this in gives us:

.

So the linear equation that goes through points A and B is:

.

Now we said earlier that the line for OP will be:

.

Opposite just means change the sign.

Reciprocal just means we are going to flip.

So the opposite reciprocal of is:

.

So the equation when graphed that goes through pts O & P is:

.

Now to actually find this point P, I need to find the intersection of the lines I have found. This lines I found for AB & OP respectively are:

I'm going to use substitution to solve this system.

To solve this equation for x, I need to get the terms that contain x on one side while the terms not containing x on the opposing side.

I'm going to distribute the to both terms in the ( ) next to it:

Now I'm to subtract on both sides:

Factor out the x on the right hand side:

Now divide both sides by what x is being multiplied by:

We need to clear the mini-fractions by multiplying top and bottom by :

Distributing the - in front of the ( ) on bottom:

I'm going to factor the out on top since both of those terms have that as a common factor:

This simplify to:

Now let's find the corresponding y-coord using either of one our equations.

I prefer the line for OP:

with

's cancel:

.

So point P is .

So now we can actually use the distance formula to compute the thing we called the height of the triangle which was the distance between O & P:

Let's simplify the top using :

.

step-by-step explanation:

true !

step-by-step explanation: