# Coordinate Proof Using Distance with Segments and Triangles

Proofs involving coordinates can be difficult to understand, especially for those who are new to the subject. Coordinate proof using distance with segments and triangles is one of many methods used to prove theorems and postulates involving geometry. By using the concept of distance, it is possible to prove that certain segments and/or triangles have certain properties. This article will go over the basics of coordinate proof using distance with segments and triangles.

## Defining Distance

Distance is a term used to describe the length of a line segment. It is the length of the path between two points in a plane. The distance between two points is the sum of the lengths of the segments that connect the two points. To calculate the distance between two points, the Pythagorean theorem can be used. The Pythagorean theorem states that the square of the length of the hypotenuse (the longest side of a right triangle) is equal to the sum of the squares of the lengths of the other two sides.

## Distance and Segments

When it comes to coordinate proof using distance with segments, one of the most common proofs involves proving that two line segments are equal in length. This can be done by using the distance formula to calculate the length of both segments and then comparing the two values. If the two values are equal, then the segments are equal in length. Another common proof involving segments is proving that a segment is congruent to the sum of two other segments. This can be done by using the distance formula to calculate the lengths of all three segments and then showing that the two sums are equal.

## Distance and Triangles

When it comes to coordinate proof using distance with triangles, one of the most common proofs involves proving that two triangles are similar. To do this, the Pythagorean theorem can be used to calculate the lengths of the sides of both triangles. Then, the ratio of the lengths of each side of the two triangles can be compared. If the ratios of all three sides match, then the triangles are similar. Another common proof involving triangles is proving that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. This can be done by using the distance formula to calculate the lengths of all three sides and then showing that the sum of any two sides is greater than the third side.

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Coordinate proof using distance with segments and triangles is a useful method for proving various theorems and postulates involving geometry. By using the concept of distance and the Pythagorean theorem, it is possible to prove that certain segments and/or triangles have certain properties. This article has gone over the basics of coordinate proof using distance with segments and triangles.