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Session 6, Part C: Applications of the Pythagorean Theorem
 
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Session 6, Part C:
Applications of the Pythagorean Theorem

In This Part: Finding Missing Lengths | The Distance Formula

"As long as algebra and geometry have been separated, their progress has been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection." -- Joseph-Louis Lagrange (1736-1813)

French mathematician René Descartes (1596-1650) was the first to employ algebra in solving geometry problems. His key insight, and one that has affected the study of mathematics ever since, was that of the development of coordinate geometry (or Cartesian geometry, named for its creator).

By convention, the axes are shown in the horizontal and vertical positions. The horizontal axis is called the x-axis, and the vertical is called the y-axis. When you describe a point, you list the coordinates in order (x,y). There's nothing magical in these conventions; they just make it easy for everyone to understand each other.

The coordinates of the point are about (2.1,1.4).

To use the power of algebra to solve problems in geometry, you need the distance formula -- a way to measure the distance between two points. Luckily, the distance formula is just a special case of the Pythagorean theorem, so you don't have to remember anything new.

Use the following Interactive Activity to experiment with the distance formula and to answer Problem C4.

This activity requires the Flash plug-in, which you can download for free from Macromedia's Web site. For a non-interactive version of this activity, plot the given pair of points on graph paper and find the distance between them by applying what you know from the Pythagorean theorem.


 

Problem C4

Solution  

Find the distances between the following:

a. 

A = (0,0) and B = (1,1)

b. 

A= (2,3) and B = (-1,-1)

c. 

A = (-3,-2) and B = (5,4)

d. 

A = (-3,4) and B (1,-1)

e. 

A= (-2,-3) and B (-3,5)

f. 

A = (-3,-1) and B (0,0)


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Try constructing a right triangle whose hypotenuse is a line segment connecting points A and B, with a as its horizontal side (parallel to the x-axis). Then use the distance formula, the algebraic version of the Pythagorean theorem.   Close Tip

 

Problem C5

Solution  

Plot the given pair of points and find the distance between them.

a. 

I = (10,-7) and J = (2,-7)

b. 

K = (1,5) and L = (1,-15)

c. 

A = (x,a) and B = (x,b)

d. 

A = (a,y) and B = (b,y)


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
In parts (c) and (d), make sure your answer takes into consideration that you don't know whether length a or b is greater. Is there a way to be sure your answer is never negative?   Close Tip

 

Problem C6

Solution  

Find the distance between the two points pictured: A=(x1,y1) and B=(x2,y2).


 

Diagram in Part C: Distance Formula adapted from Connected Geometry, developed by Educational Development Center, Inc. p. 355. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

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