A Point Q on a Segment with Endpoints A and C

Oct 13, 2025 by windailys.com 107 views

A Point Q on a Segment with Endpoints A and C

In this article, we will explore the concept of partitioning a line segment into a specific ratio. We will use the given endpoints A (2, -1) and C (4, 2) to find the coordinates of point Q, which partitions the segment in a 4:1 ratio.

To solve this problem, we need to understand the concept of partitioning a line segment. When a line segment is partitioned in a specific ratio, it means that the segment is divided into two parts, with the first part being a certain fraction of the total length and the second part being the remaining fraction.

To find the coordinates of point Q, we need to use the concept of similar triangles. We can draw a line from point A to point C and then draw a line from point A to point Q. The line from point A to point Q will be a fraction of the total length of the line from point A to point C.

Step 1: Find the Total Length of the Line Segment

The total length of the line segment can be found using the distance formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In this case, the coordinates of points A and C are (2, -1) and (4, 2), respectively. Plugging these values into the distance formula, we get:

d = \sqrt{(4 - 2)^2 + (2 - (-1))^2}
d = \sqrt{2^2 + 3^2}
d = \sqrt{4 + 9}
d = \sqrt{13}

Step 2: Find the Length of the First Part of the Line Segment

Since the line segment is partitioned in a 4:1 ratio, the first part of the line segment will be 4/5 of the total length. We can find the length of the first part by multiplying the total length by 4/5:

l_1 = \frac{4}{5}d
l_1 = \frac{4}{5}\sqrt{13}

Step 3: Find the Coordinates of Point Q

We can use the following formula to find the coordinates of point Q:

x_Q = x_A + \frac{l_1}{d}(x_C - x_A)
y_Q = y_A + \frac{l_1}{d}(y_C - y_A)

Plugging in the values, we get:

x_Q = 2 + \frac{\frac{4}{5}\sqrt{13}}{\sqrt{13}}(4 - 2)
x_Q = 2 + \frac{4}{5}(2)
x_Q = 2 + \frac{8}{5}
x_Q = \frac{10}{5} + \frac{8}{5}
x_Q = \frac{18}{5}

y_Q = -1 + \frac{\frac{4}{5}\sqrt{13}}{\sqrt{13}}(2 - (-1))
y_Q = -1 + \frac{4}{5}(3)
y_Q = -1 + \frac{12}{5}
y_Q = \frac{-5}{5} + \frac{12}{5}
y_Q = \frac{7}{5}

In this article, we used the concept of partitioning a line segment to find the coordinates of point Q. We found that the coordinates of point Q are (18/5, 7/5).

The final answer is (18/5, 7/5).
A Point Q on a Segment with Endpoints A and C: Q&A

In our previous article, we explored the concept of partitioning a line segment into a specific ratio. We used the given endpoints A (2, -1) and C (4, 2) to find the coordinates of point Q, which partitions the segment in a 4:1 ratio. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the concept of partitioning a line segment?

A: Partitioning a line segment means dividing it into two parts, with the first part being a certain fraction of the total length and the second part being the remaining fraction.

Q: How do you find the coordinates of point Q?

A: To find the coordinates of point Q, you need to use the concept of similar triangles. You can draw a line from point A to point C and then draw a line from point A to point Q. The line from point A to point Q will be a fraction of the total length of the line from point A to point C.

Q: What is the formula to find the coordinates of point Q?

A: The formula to find the coordinates of point Q is:

x_Q = x_A + \frac{l_1}{d}(x_C - x_A)
y_Q = y_A + \frac{l_1}{d}(y_C - y_A)

Q: What is the significance of the 4:1 ratio in this problem?

A: The 4:1 ratio means that the first part of the line segment will be 4/5 of the total length, and the second part will be 1/5 of the total length.

Q: How do you find the length of the first part of the line segment?

A: To find the length of the first part of the line segment, you need to multiply the total length by 4/5:

l_1 = \frac{4}{5}d

Q: What is the total length of the line segment?

A: The total length of the line segment can be found using the distance formula:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Q: Can you explain the concept of similar triangles in this problem?

A: Yes, similar triangles are used to find the coordinates of point Q. The line from point A to point Q is a fraction of the total length of the line from point A to point C. This is because the two triangles formed by the lines are similar.

Q: How do you use the concept of similar triangles to find the coordinates of point Q?

A: To use the concept of similar triangles, you need to draw a line from point A to point C and then draw a line from point A to point Q. The line from point A to point Q will be a fraction of the total length of the line from point A to point C. You can then use the formula to find the coordinates of point Q.

In this article, we answered some frequently asked questions related to the topic of partitioning a line segment. We hope that this article has provided you with a better understanding of the concept and how to find the coordinates of point Q.

The final answer is (18/5, 7/5).