Distance
The distance between two points on a number line is the absolute value of the difference between the coordinates. If the coordinates of the points \(A\) and \(B\) are \(a\) and \(b\), then the distance between them is \(|a - b|\) or \(|b - a|\). The distance between \(A\) and \(B\) is also called the length of \(\overline{AB}\) or \(AB\).
The distance between two points on a number line is the absolute value of the difference between the coordinates. If the coordinates of the points \(A\) and \(B\) are \(a\) and \(b\), then the distance between them is \(|a - b|\) or \(|b - a|\). The distance between \(A\) and \(B\) is also called the length of \(\overline{AB}\) or \(AB\).
Example 1:
Find the length of the segment below.
Find the length of the segment below.
Solution:
Length = \(|3 - (-5)| = |8| = 8\) units
Length = \(|3 - (-5)| = |8| = 8\) units
Segment Addition Postulate
If three points are colinear and \(B\) is between \(A\) and \(C\), then \(AB + BC = AC\).
If three points are colinear and \(B\) is between \(A\) and \(C\), then \(AB + BC = AC\).
What if the coordinates for the two points are not on a number line? We will need to use the Distance Formula.
The Distance Formula is based on the Pythagorean Theorem.
Distance Formula
\((AB)^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2\) |
Pythagorean Theorem
\(c^2 = a^2 + b^2\) |
Example 3:
Find the length of the segment. Round to the nearest tenth of a unit.
Find the length of the segment. Round to the nearest tenth of a unit.
Solution:
Using the Distance Formula
\(\begin{align*}
MS &= \sqrt{(2- -3)^2 + (-1 - 3)^2}\\
&=\sqrt{(5)^2 + (-4)^2}\\
&=\sqrt{25 + 16}\\
&=\sqrt{41} \approx6.4
\end{align*}\)
OR
Using the Pythagorean Theorem
Using the Distance Formula
\(\begin{align*}
MS &= \sqrt{(2- -3)^2 + (-1 - 3)^2}\\
&=\sqrt{(5)^2 + (-4)^2}\\
&=\sqrt{25 + 16}\\
&=\sqrt{41} \approx6.4
\end{align*}\)
OR
Using the Pythagorean Theorem
Midpoint
The midpoint of a segment is the point that divides or bisects the segment into two congruent segments. To bisect a segment means to cut it into two equal parts. The word congruent means to be the same size and shape. Congruent segments have the same length. Congruent angles are the same size or same angle measure.
If \(M\) is the midpoint of \(\overline{AB}\), then \(AM = MB\). We are using red tick marks to indicate congruent parts in these diagrams.
The midpoint of a segment is the point that divides or bisects the segment into two congruent segments. To bisect a segment means to cut it into two equal parts. The word congruent means to be the same size and shape. Congruent segments have the same length. Congruent angles are the same size or same angle measure.
If \(M\) is the midpoint of \(\overline{AB}\), then \(AM = MB\). We are using red tick marks to indicate congruent parts in these diagrams.
A segment bisector is a point, ray, line segment or plane that intersects a line segment at its midpoint. A midpoint or segment bisector bisects a segment
In the diagram above, \(\overrightarrow{ST}\) bisects \(\overline{AB}\) because \(\overline{AT}\cong\overline{TB}\)
Midpoint Formula
The coordinates of the midpoint of a segment are the averages of the x-coordinates and the y-coordinates of the endpoints of the segments.
If \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the endpoints of the segment, then the midpoint, \(M\), has the coordinates
\(M = \left(\Large\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
The coordinates of the midpoint of a segment are the averages of the x-coordinates and the y-coordinates of the endpoints of the segments.
If \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are the endpoints of the segment, then the midpoint, \(M\), has the coordinates
\(M = \left(\Large\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
Example 4:
Find the coordinates of the midpoint of the segment with the given endpoints \(G(-7, -5)\) and \(T(-1, 9)\).
Solution:
Using the Midpoint Formula:
\(\left(\Large\frac{-7 + -1}{2}, \frac{-5 + 9}{2}\right) = \left(\Large\frac{-8}{2}, \frac{4}{2}\right) = (-4, 2)\)
Example 5:
Line \(t\) bisects the segment. Find \(RW\) if \(RT = 7\frac{1}{4}\) in.
Solution:
\(RT = 7\frac{1}{4}\) in
\(RW = RT\cdot2 = 7\frac{1}{4}\cdot(2) = 7.25(2)= 14.5\) in.
Example 6:
Use the given endpoint \(H\) and midpoint \(M\) of \(\overline{HJ}\) to find the coordinates of the other endpoint \(J\). \(H(5, 1)\) and \(M(1, 3)\)
\(RT = 7\frac{1}{4}\) in
\(RW = RT\cdot2 = 7\frac{1}{4}\cdot(2) = 7.25(2)= 14.5\) in.
Example 6:
Use the given endpoint \(H\) and midpoint \(M\) of \(\overline{HJ}\) to find the coordinates of the other endpoint \(J\). \(H(5, 1)\) and \(M(1, 3)\)
Solution:
Let's watch the video for the solution.
Let's watch the video for the solution.
Quick Check
1) Determine the length of the segment below:
1) Determine the length of the segment below:
2) If \(N\) is between \(O\) and \(E\), where \(ON = 2x - 5\), \(NE = 4x + 9\), find \(OE\).
3) Determine the length of the segment. Round to the nearest tenth of a unit.
3) Determine the length of the segment. Round to the nearest tenth of a unit.
4) Find the coordinates of the midpoint of the segment with the given endpoints \(R(-6, 4)\) and \(S(2, 5)\).
5) Line \(t\) bisects \(\overline{CT}\). If \(CA = 2x - 1\) and \(AT = 5x - 7\), determine \(x\).
5) Line \(t\) bisects \(\overline{CT}\). If \(CA = 2x - 1\) and \(AT = 5x - 7\), determine \(x\).
6) Use the given endpoint \(D\) and midpoint \(M\) of to find the coordinates of the other endpoint \(E\). \(D(4, 5)\) and \(M(0, 2)\).