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 :
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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:
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So the linear equation that goes through points A and B is:
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Now we said earlier that the line for OP will be:
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Opposite just means change the sign.
Reciprocal just means we are going to flip.
So the opposite reciprocal of is:
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So the equation when graphed that goes through pts O & P is:
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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
Â
's cancel:
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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 :
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