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Midpoint of a Line Segment
of a line segment passing through two points.
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Let $C_1$ be a circle defined with $X(2,7)$ and $Y(2,5)$ as the endpoints of the diameter of the circle.
Let $C_2$ be a circle defined with $Y(2,5)$ and $Z(4,11)$ as the endpoints of the diameter of the circle.
Let $A = C_1 \cup C_2 – Y$
Let $B$ be the set of all midpoints of the set of all straight line segments joining two distinct points in $A$ and passing through the point $Y$. Find $B$.
My attempt:
Since $XYZ$ are collinear $C_1$ and $C_2$ meet at a single point: $Y$. I can easily find the set of all midpoints of the line segments joining a point (say $T$) in $C_1$ and $Y$. Let $M(x,y)$ be the midpoint of $TY$ and $N(0,1)$ be the center of $C_1$. Notice that $MN$ is perpendicular to to $MY$. Therefore we can build the equation $\dfrac{y1}{x0}*\dfrac{y+5}{x2}=1$ i.e product of their slopes equal 1. Similarly, the set of all midpoints of the line segments joining a point (say $U$) in $C_2$ and $Y$ can be found.
The issue now is, I don’t know how to use this information to compute the required locus $B$. I don’t even know if this information is useful.
nonuser
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asked Jan 5, 2021 at 7:25
SidSid
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Notice that transformation $A\mapsto B$ from bigger circle which has radius $2\sqrt{10}$: $$XY = \sqrt{4^2 +12^2} = 4\sqrt{10}$$ and center at $O(0,1)$ to smaller circle with radius $\sqrt{10}$ and center at $O'(3,8)$ is homothety with center at $Y$ and dilatation factor $1/2$. So, for all $A$ we have $${BY\over YA} = {1\over 2}\implies {AB\over YA} ={3\over 2}$$
Then $${YM \over YA} = 1 {AM\over YA} = 1{AB\over 2YA} = {1\over 4} $$
So $M$ is actually a picture of $A$ under homothety at $Y$ with dilatation factor $1/4$, so $M$ descibes a circle with radius $r=\sqrt{10}/2$ and center at $({1\over 2},{7\over 2})$.
answered Jan 5, 2021 at 7:59
nonusernonuser
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Locus of points dividing segments joining points of $S_1$ with points of $S_2$ in ratio $\lambda : (1\lambda)$ is given by $$\lambda S_1 + (1\lambda) S_2=0$$
$(2,5)$ should lie on this locus, which it clearly does. For midpoints, put $\lambda = \frac{1}{2}$. The coefficient of $x^2$ and $y^2$ will turn out to be same; this locus is a circle.
Given $$S_1 : (x2)(x+2) + (y+5)(y7)=0$$
$$S_2 : (x2)(x4) + (y+5)(y+11)=0$$
Required locus is $$S : (x2)(x1) + (y+5)(y+2)=0$$
with center at $$\Big(\dfrac{2+1}{2},\dfrac{52}{2}\Big)=\Big(\dfrac{3}{2},\dfrac{7}{2}\Big)$$
answered Jan 5, 2021 at 10:38
cosmo5cosmo5
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Frequently Asked Questions About can a line segment have two midpoints explain
If you have questions that need to be answered about the topic can a line segment have two midpoints explain, then this section may help you solve it.
Can there be two midpoints on a line segment?
In geometry, a line segment is a line with two endpoints; the midpoint of a line segment is the point on the line segment that is the same distance from each of the line segment’s endpoints.
The two segments on either side of a midpoint are equal to one another for what reason?
Each segment between the midpoint and an endpoint will be the same length because a segment midpoint divides a line segment into two congruent segments.
A segment may have up to three points.
There are infinitely many points on a line segment, including points between the line’s endpoints, between its middle point and its start, and between its endpoints, and there are more than two points on a line segment.
Can the midpoint be the only one?
Because of this, it is incorrect to assume that a line segment AB has two midpoints; instead, each line segment will only have one specific midpoint.
Can you succinctly describe what a line segment’s midpoint is?
The point on a line segment that divides it into two congruent segments is known as the midpoint of the segment, which is also known as the point on the segment that is halfway between the two ending points.
A line’s midpoint may exist.
The midpoint is located halfway between the two end points; its x and y values are in the middle of the two x and y values, respectively.
The midpoint theorem: Why is it?
The midpoint theorem asserts that “the line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half the length of the third side” (emphasis added).
Does a midpoint imply equality on both sides?
In a triangle, the segment formed by connecting the midpoints of two sides must be parallel to the third side and also half the length of the third side. The midpoint theorem states that in a triangle, the segment formed by connecting the midpoints of two sides must be half the length of the third side.
Is a line segment made up of only two points?
A line segment in geometry is a portion of a line that connects two points; in contrast to a line, which has no endpoints and can extend indefinitely in both directions, a line segment has two fixed or definite endpoints.
Can a line segment have two directions?
A ray is a line segment that extends indefinitely in one direction, a line segment is a portion of a line that has two endpoints and is finite in length, and a line is a straight path on a plane that extends forever in both directions with no endpoints.
How can it be demonstrated that a line segment has a single midpoint?
Let’s say AB is a line segment, and C is its midpoint; if D is an additional midpoint of AB, AD=DB; C is AB’s midpoint; Account=CB; therefore, ACAD=CBDB; DC=DC; DC+DC=0 2DC=0; DC=02 DC=0; and, consequently, C and D coincide. Consequently, every line segment has one and only one midpoint.
Does each line segment have a single midpoint?
It is true that there is exactly one midpoint for each line segment and that there is exactly one angle bisector for each angle.
Is it possible for the middle of a line to provide justification for your response?
However, a line cannot have a midpoint because it lacks end points.
Why is there no midpoint on a line?
We are aware that the line is infinitely long and that the midpoint cannot be located.
A line segment is defined by what?
The length of a line segment can be measured in either metric units like millimeters or centimeters or in conventional units like feet or inches. Accordingly, a line segment is a piece or part of a line having two endpoints.
In a line, how many middle points are there?
The line connecting the two reference points, which are the line’s endpoints, is divided into two equal halves by the midpoint, which is located in the middle of the line.