MathAdmin at 23:49, 4 October 2015

2015-10-04T23:49:24Z

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==Introduction== The Distance and midpoint formula allow us to talk about the simplest interesting geometric objects, lines. There is a sa..."

2015-10-04T22:55:14Z

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==Introduction==
The Distance and midpoint formula allow us to talk about the simplest interesting geometric objects, lines. There is a saying that the shortest path between two points is a line, but without a way to measure distance
we are unable to determine the shortest distance. The midpoint of a line gives some insight into more geometric concepts.

==Distance Formula==
Given two points P and Q with coordinates <math> (x_1, y_1)\text{ and }(x_2, y_2)</math>, respectively. One way to think about the distance between P and Q is to draw the line segment between P and Q, and find a right
triangle where the line segment PQ is the hypotenuse.

Example:
Find the distance between (1, 3) and (5, 6).

Solution:
In order to form the right triangle we notice that to get from (1, 3) to (5, 6) we could travel 4 units to the right then 3 units up. So you could travel from (1, 3) to (5, 3) then finally to (5, 6).
Thus, we have created a triangle with vertices (1, 3), (5, 6), and (5, 3). We also have the additional property that the hypotenuse is the segment between (1, 3) and (5, 6). The side lengths of this triangle
are 3, 4 and some unknown value for the hypotenuse. Thus, the distance from (1, 3) to (5, 6) is <math>\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5. </math>

==Midpoint Formula==
The midpoint between two points P and Q is the point on the line segment PQ that is halfway between P and Q. The formula for the midpoint is <math>\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)</math>, where the coordinates
of P are <math>(x_1, y_1)</math> and the coordinates of Q are <math> (x_2, y_2)</math>

Example:

Find the midpoint of the line segment between P(-1, 5) and Q( 4, 3)

Solution. Using the formula the midpoint is <math>\left(\frac{-1 + 4}{2}, \frac{5 + 3}{2}\right) = \left(\frac{3}{2}, 4\right)

[[Math_5|'''Return to Topics Page]]

@@ Line 25: / Line 25: @@
 Find the midpoint of the line segment between P(-1, 5) and Q( 4, 3)
-Solution. Using the formula the midpoint is <math>\left(\frac{-1 + 4}{2}, \frac{5 + 3}{2}\right) = \left(\frac{3}{2}, 4\right)
+Solution. Using the formula the midpoint is <math>\left(\frac{-1 + 4}{2}, \frac{5 + 3}{2}\right) = \left(\frac{3}{2}, 4\right)</math>
 [[Math_5|'''Return to Topics Page]]

Distance and Midpoint Formulas - Revision history

MathAdmin at 23:49, 4 October 2015

MathAdmin: Created page with "__TOC__ ==Introduction== The Distance and midpoint formula allow us to talk about the simplest interesting geometric objects, lines. There is a sa..."

MathAdmin: Created page with "
TOC
==Introduction== The Distance and midpoint formula allow us to talk about the simplest interesting geometric objects, lines. There is a sa..."