If Point { (1,2)$}$ Is The Midpoint Of The Line Segment Joining The Points { (3,5)$}$ And { (2a, B)$}$, Then { (a, B) =$}$A. { (-1,-1)$}$ B. { \left(-\frac{1}{2},-\frac{1}{2}\right)$}$ C.
If Point (1,2) is the Midpoint of the Line Segment Joining the Points (3,5) and (2a, b), Then (a, b) = ?
Understanding Midpoint Formula
The midpoint formula is a fundamental concept in mathematics that helps us find the midpoint of a line segment given the coordinates of its endpoints. The midpoint formula is given by:
(x1 + x2)/2, (y1 + y2)/2)
where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the line segment.
Given Information
We are given that the point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b). Using the midpoint formula, we can write:
(3 + 2a)/2, (5 + b)/2 = (1, 2)
Solving for a and b
To solve for a and b, we can equate the x-coordinates and y-coordinates separately.
Equating X-Coordinates
(3 + 2a)/2 = 1
Multiplying both sides by 2, we get:
3 + 2a = 2
Subtracting 3 from both sides, we get:
2a = -1
Dividing both sides by 2, we get:
a = -1/2
Equating Y-Coordinates
(5 + b)/2 = 2
Multiplying both sides by 2, we get:
5 + b = 4
Subtracting 5 from both sides, we get:
b = -1
Conclusion
Therefore, we have found that a = -1/2 and b = -1.
Answer
The correct answer is:
- B.
Explanation
The point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b). Using the midpoint formula, we can write:
(3 + 2a)/2, (5 + b)/2 = (1, 2)
Equating the x-coordinates, we get:
(3 + 2a)/2 = 1
Multiplying both sides by 2, we get:
3 + 2a = 2
Subtracting 3 from both sides, we get:
2a = -1
Dividing both sides by 2, we get:
a = -1/2
Equating the y-coordinates, we get:
(5 + b)/2 = 2
Multiplying both sides by 2, we get:
5 + b = 4
Subtracting 5 from both sides, we get:
b = -1
Therefore, we have found that a = -1/2 and b = -1.
Final Answer
The final answer is .
Q&A: If Point (1,2) is the Midpoint of the Line Segment Joining the Points (3,5) and (2a, b), Then (a, b) = ?
Frequently Asked Questions
Q: What is the midpoint formula?
A: The midpoint formula is a fundamental concept in mathematics that helps us find the midpoint of a line segment given the coordinates of its endpoints. The midpoint formula is given by:
(x1 + x2)/2, (y1 + y2)/2)
Q: How do we find the midpoint of a line segment?
A: To find the midpoint of a line segment, we can use the midpoint formula. We simply add the x-coordinates and y-coordinates of the endpoints and divide by 2.
Q: What is the given information in this problem?
A: The given information is that the point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b).
Q: How do we solve for a and b?
A: To solve for a and b, we can equate the x-coordinates and y-coordinates separately. We can use the midpoint formula to set up equations and then solve for a and b.
Q: What are the values of a and b?
A: The values of a and b are a = -1/2 and b = -1.
Q: What is the correct answer?
A: The correct answer is:
- B.
Q: Why is the point (1, 2) the midpoint of the line segment joining the points (3, 5) and (2a, b)?
A: The point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b) because it satisfies the midpoint formula. When we substitute the values of the endpoints into the midpoint formula, we get:
(3 + 2a)/2, (5 + b)/2 = (1, 2)
Q: How do we know that the point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b)?
A: We know that the point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b) because it satisfies the midpoint formula. When we equate the x-coordinates and y-coordinates separately, we get:
(3 + 2a)/2 = 1
(5 + b)/2 = 2
Q: What is the final answer?
A: The final answer is .
Conclusion
In this article, we have discussed the problem of finding the values of a and b given that the point (1, 2) is the midpoint of the line segment joining the points (3, 5) and (2a, b). We have used the midpoint formula to set up equations and then solved for a and b. The correct answer is .