Algebra 2 Geometric Word Problems
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On The Inside
Geometric Word Problems

 

 What is the largest rectangle that can be drawn inside a right triangle

 with sides of 9 and 12 units and hypotenuse of 15 units?
 
 Before you do anything else, draw a picture of the problem.

 

 
 OK, the way to solve a problem like this
 is to place it on an X, Y graph.
 Put the right triangle at the origin (0,0) point.

 

 The trick to solving this one is to use the equation of the line
 that is the hypotenuse of the triangle.
 We know two points on that line: (0,9) and (12,0).
 With these two points, we can find the equation of the line.
 First we need to find the slope of the line.
 The line goes down from left to right 
 so the slope will be a negative number.
 

 

 The other thing we need to know 
 is the point where the line crosses the Y axis.
 That is, the point where X = 0.
 We don't need to look too hard for that point, It's at (0,9)!
 
 The slope intercept form of the equation of a line looks like this:
 

 Y = mX + b

 
 And for our line, b = 9 and m = -3/4.
 That means the equation of the line is:
 

 

 
 This is the equation of the hypotenuse line.
 Every point on the hypotenuse has an X, Y value
 that makes the equation true.
 THAT INCLUDES THE POINT THAT IS THE CORNER OF THE RECTANGLE!!!
 
 We can call that point, whatever it is, (X, Y)
 But we also know that the point is on the line:
 

 

 
 So instead of using the name Y, and calling the point (X, Y),
 We can call it:

 

 
 We can also name the points on the other three corners of the rectangle:

 

 
 Now the last trick.
 From left to right, 
 the rectangle goes from an X value of 0 to a value of X.
 So the width of the rectangle is X.
 From bottom to top,
 the rectangle goes from a value of Y = 0 to Y = -3/4X + 9.
 So the height of the rectangle is -3/4X + 9
 
 THAT WAS THE BIG TRICK OF THIS PAGE.
 DID YOU CATCH IT?
 WE WERE ABLE TO WRITE Y IN TERMS OF X
 

 

 So the area equation is:

 

 
 Well what do you know? An upside down "U"!
 

 

 So the maximum area happens when X = 6, What is Y?
 

 

 So:

 

 

   copyright 2005 Bruce Kirkpatrick

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