How to compare two segments : an entertaining geometry

You will need:
• Compass
• line
• ability to carry out elementary arithmetic
# 1

Compare two segments - which means to determine the length of which of them is more or less relative to the other.In the real world, such operations produce many of us, without even noticing.We compare the length of the roads on the map to choose a shorter route, determine the higher of the brothers, meryaya and comparing their growth, and on the line or plant comparison of the lengths of similar values ​​applied often.Our task - to be able to build a mathematical model for every application, be able to solve it properly.It is also possible to face or improvised tools to compare the two segments.Let us assume that a greater length: match or the cap of a ballpoint pen?Caliper measurements of the length of the match and attach it to the cap we can immediately get the answer.

# 2

But how to compare the two segments, if their length is indistinguishable to the eye?If you are unable to use the means at hand, but

we are only just the coordinates of the segment?In the case of one-dimensional space, you can compare the two segments by finding their lengths.On the line length of the segment - is the difference between the coordinate values ​​of his day, taken with the plus sign.For example: given a segment AB with coordinates A (2), B (3) and the segment CD with coordinates C (5,1) and D (6).Determine which of the segments is long.The length of AB is equal to 2.3 = 1, and CD will be equal to the length 6-5,1 = 0.9.s It follows that the segment AB larger than a CD.Consider another task.Given the coordinates of the segment KL: 0 and 4, respectively.Also beginning coordinates given segment MN M (-3) and coordinate of the middle of this segment (1).Compare the length of KL and MN segments.

# 3

To solve this problem it is necessary to know how to find the coordinates of the midpoint.Coordinate of the midpoint is the arithmetic mean of the coordinates of its ends.For our problem, it turns out that the coordinate M (-3) plus an unknown coordinate N (x) by dividing in half give -1.Composition and solve the equation.(-3 + X) / 2 = -1.We multiply both sides by -2 -3 + x = -2.Fast forward 3 in the right-hand side of the equation, changing the sign: x = 1.We find that the coordinate of N is equal to 1. We find the length of the segment MN: 1 - (- 3) = 1 + 3 = 4.Similarly, the length KL = 4-0 = 4. As can be seen, the length of the segments are the same, therefore the segments are equal.

# 4

for geometric problems is often important to know the name of the segment connecting these or those two points.Sometimes it helps to avoid solving the problem in a general way and apply the theorem and simplified method of solving the problem.However, solve the problem which applies a common formula for finding the length of a segment of the coordinates of its ends.For the plane length of the segment is equal to the square root of the sum of the squares of the differences between corresponding coordinates of its ends.This formula is a generalization of the one-dimensional space, that is.in turn, it is a special case of formula for the three-dimensional, and so on.Knowing how to use these formulas, we can find the length of the segment in the plane and in space.Let us turn directly to the problem.

# 5

task.The point with coordinates (-3, 2) is a common beginning CB and CA intervals.Point A has coordinates (0, 0) and point B - (1, 4).Compare segments CB and CA.Decision.We calculate the length of the segment CA on the above formula: root of -3-0 = -3 in the box, this value is equal to 9,2-0 = 2, erecting two in the square obtain 4. The sum of the squares of the differences is equal to 13, therefore the length of the CA is the root of13. Applying similar arithmetic to find the length of the NE we find that the length of this segment is -3-1 = -4.-4 * -4 = 6.2-4 = -2.-2 * -2 = 4.6 + 4 = 20, hence the length of the segment is equal to the square root of the SW 20. The square root of 20 is greater than the square root of 13, hence the longer segment CB CA segment.The problem is solved.