If r is the midpoint of qs rs 2x 4 – If R is the midpoint of QS and RS = 2x + 4, we embark on a geometric exploration to unravel the relationship between the coordinates of R, Q, and S. Delving into the properties of midpoints, we will derive an equation that guides us to the coordinates of R, solidifying our understanding of this fundamental concept.
Introduction: If R Is The Midpoint Of Qs Rs 2x 4
In geometry, the midpoint of a line segment is the point that divides the segment into two equal parts. In other words, it is the center point of the segment.
We are given that R is the midpoint of QS and RS = 2x + 4.
Properties of a Midpoint
- The midpoint divides the line segment into two equal parts.
- The midpoint is equidistant from the endpoints of the line segment.
- The midpoint is located on the line segment.
Finding the Coordinates of the Midpoint
The coordinates of the midpoint of a line segment can be found using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.
Properties of Midpoints
Definition of Midpoint, If r is the midpoint of qs rs 2x 4
A midpoint is a point that divides a line segment into two equal parts. It is the central point of the line segment, equidistant from both endpoints.
Relationship Between Midpoint Coordinates and Endpoint Coordinates
The coordinates of a midpoint can be calculated using the average of the coordinates of the endpoints it connects. If the endpoints are denoted by (x1, y1) and (x2, y2), then the coordinates of the midpoint M are given by:
M = ((x1 + x2) / 2, (y1 + y2) / 2)
This formula can be used to find the midpoint of any line segment.
Application to the Given Information
The midpoint formula, which states that the midpoint of a line segment connecting two points is the average of their coordinates, can be applied to the given information to derive an equation relating the coordinates of R, Q, and S.
Since R is the midpoint of QS, the coordinates of R are:
R = ((x1 + x2)/2, (y1 + y2)/2)
where (x1, y1) are the coordinates of Q and (x2, y2) are the coordinates of S.
Substituting the given values, we get:
R = ((2 + 4)/2, (3 + 5)/2) = (3, 4)
Therefore, the coordinates of R are (3, 4).
Verification
To confirm that R is indeed the midpoint of QS, we need to verify that it satisfies the midpoint formula. This involves substituting the coordinates of R back into the given equation and checking if it holds true.
The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by: $$\left(\fracx_1 + x_22, \fracy_1 + y_22\right)$$
Substituting the coordinates of Q (2, 4) and S (6, 8) into the midpoint formula, we get:
$$\left(\frac2 + 62, \frac4 + 82\right) = \left(\frac82, \frac122\right) = (4, 6)$$
These coordinates match the coordinates of R, which means that R satisfies the midpoint formula and is therefore the midpoint of QS.
Example
Let’s consider a numerical example to illustrate the concept of finding the coordinates of the midpoint R when the coordinates of Q and S are given.
Suppose we have two points Q(2, 4) and S(6, 8). We can use the midpoint formula to find the coordinates of R, which is the midpoint of QS.
Finding the Coordinates of R
- Substitute the coordinates of Q and S into the midpoint formula:
- Plug in the values:
- Simplify:
$$R = (\fracx_Q + x_S2, \fracy_Q + y_S2)$$
$$R = (\frac2 + 62, \frac4 + 82)$$
$$R = (4, 6)$$
Therefore, the coordinates of the midpoint R of the line segment QS are (4, 6).
FAQ Overview
What is the definition of a midpoint?
A midpoint is a point that divides a line segment into two equal parts.
How do you find the coordinates of a midpoint?
The coordinates of a midpoint are the average of the coordinates of the endpoints it connects.
What is the significance of verifying that a point is a midpoint?
Verifying that a point is a midpoint ensures that it satisfies the definition of a midpoint and correctly divides the line segment into two equal parts.
