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Who likes mountaineering? I recall, as part of our induction into college we the newbies had to trek long distances including mountain climbing. In order to prevent us from fainting along the journey, this long journey was split into several stop distances which were called "mountaineering segments". This article will explore everything you should know aboutsegment length.
The distance between two points on a line segment is the segment length.
It is a very concise definition. For short, segment length is all about "from one point to another". Recall the mountaineering segments, those were just parts of the total distance that we had to cover.
An illustration of line segments with a road path, StudySmarter Originals
Meanwhile, one cannot understand segment length without taking into perspective points, because these are your focus on locating where the segment starts as well as where it stops.
两个点之间的线段的长度distance between two points. One point serves as the starting point which is the location from which the measurement begins. Meanwhile, the other is the endpoint which is the location where the measurement stops. It is sometimes a name with a small letter or to letters in upper cases. For instance, if there are two points A and B, we can call the length segment existing between A and B, or c, or we may just call the line segment, AB.
An image of a segment length, StudySmarter Originals
Since we are dealing with points, we need to know their position on the cartesian plane. In other words, we should know the position of the starting and ending point on the x and y-axis. This position is referred to as the coordinates of the points of the length segment and they are written in the form (x1, y1) and (x2, y2).
Here,
x1means the position of the starting point in the x-axis,
y1means the position of the starting point in the y-axis,
x2means the position of the ending point in the x-axis
and
y2means the position of the ending point in the y-axis.
The image below describes this clearly.
A graphical image of two points of a segment length with its coordinates, StudySmarter Originals
Now we can see the length segment as not only a distance, but we now consider the points that determine this distance.
You should now think of how to determine the length of segments when you know their starting and ending points.
To find the segment length, we can create a right-angled triangle and hence use Pythagoras' theorem to solve for the distance:
An image shows the use ofPythagoras Theoremto calculate the length of a segment from two points, StudySmarter Originals
We can see that
is the vertical distance between points A and B.
is the horizontal distance between points A and B. Hence, we can complete a Pythagorean triangle by inserting the distance between points A and B as
.
Using Pythagoras' theorem, we know:
Since the distance between two points cannot be negative, we know:
For points A=(
) and B=(
):
and
Therefore:
Note that since ∆y and ∆x are both squared, there is no need to take the absolute value of these numbers as squaring them turns them positive. Also, note that the square root does not cancel the fact that ∆y and ∆x are squared as the equation adds these terms and does not multiply them.
Find the distance between the points A=(5, 0) and B=(3,7)
Solution:
Substituting the coordinates into the equation for segment length:
Unless specified, you may choose to leave your answer in exact or numerical form.
In some cases, you may be given only the endpoints and midpoints and you would need to determine the length of the whole segment.
The midpoint is the point halfway through the distance between the starting and the ending point.
When this occurs, the first step to follow is to find the starting point which was not given initially. So, for a starting point A(x1, y2), midpoint M (xm, ym) and endpoint B (x2, y2), the midpoint for x-axis is calculated as:
and the midpoint for the y-axis is calculated as:
However, our interest is in finding the starting point when only the endpoint and the midpoint are given. In this case, you just need to make in their respective cases x1or y1the subject of the formula. This means that the coordinate of the starting point in the x-axis, x1is:
Solved as
and the coordinate of the starting point in the y-axis, y1is:
Solved as
If Nonso is on a journey in which his path is linear and he has currently covered half of the distance. If his current coordinate is (4, -2) and his journey terminates at K (9, 5), find the segment length of the whole journey.
Solution:
From the information given, Nonso is currently at the midpoint of the total journey which is the segment length of the journey. Since K is where the journey ends that means we have our endpoint. With these, we can now find the coordinates of our starting point as
and
This means that Nonso started his journey at the point (-1,-9).
Now we know his starting point, we can calculate the length of the segment for the journey as:
A segment of a circle is bounded by an arc and a chord. The line segment of a circle can either be the diameter of a circle when the line passes through the center of the circle or a chord if the line passes any other place apart from the center of a circle.
To calculate the length of the segment of a circle when it passes the center, multiply the given radius by 2. However, when it passes outside the center then the length of a segment of a circle is the length of the chord calculated as
Where r is the radius and θ is the angle subtended by the sector that forms the segment.
This formula was gotten from the image description below;
An image illustrating the derivation of the length of a segment of a circle, StudySmarter Originals
from the image using SOHCAHTOA we get
Find the length of the line segment of a circle with a radius of 10 cm which subtends 120° at the center.
Solution:
Example of line segment in a circle, StudySmarter Originals
The length of the line segment is
It is the distance between two points on a straight line.
It originates from Pythagoras' theorem.
It is the area within a circle bound by a chord.
Length of segments can be found by simply applying the distance formula when the endpoints are known.
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