Finding Coordinates Point Dividing Line Segment In Given Ratio

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In geometry, a fundamental concept involves finding the coordinates of a point that divides a line segment in a given ratio. This problem explores the scenario where point R divides the line segment EF in the ratio 1:5. Given the coordinates of points E and F, we will determine the coordinates of point R using the section formula. Understanding this concept is crucial for various applications in mathematics, physics, and engineering, where precise positioning and division of segments are essential.

Point R divides the line segment EF in the ratio 1:5. The coordinates of point E are (4, 8), and the coordinates of point F are (11, 4). Our objective is to find the coordinates of point R accurate to two decimal places. This problem requires a clear understanding of the section formula, which is the cornerstone for solving such coordinate geometry problems. The correct application of this formula ensures the accurate calculation of the coordinates of the dividing point.

The section formula is a vital tool in coordinate geometry that helps determine the coordinates of a point that divides a line segment internally in a given ratio. If a point R(x, y) divides the line segment joining points E(x₁, y₁) and F(x₂, y₂) in the ratio m:n, then the coordinates of R can be calculated using the following formulas:

Formula Explanation

  • x-coordinate of R: x = (m * x₂ + n * x₁) / (m + n)
  • y-coordinate of R: y = (m * y₂ + n * y₁) / (m + n)

Variables Definition

  • m and n represent the ratio in which the point R divides the line segment EF.
  • (x₁, y₁) are the coordinates of point E.
  • (x₂, y₂) are the coordinates of point F.

Understanding and correctly applying this formula is the key to solving problems involving the division of line segments. The section formula provides a straightforward method to find the exact location of a point along a line segment given the ratio of division.

Applying the Section Formula to Find Coordinates of Point R

Given that point R divides line segment EF in the ratio 1:5, with E(4, 8) and F(11, 4), we can now apply the section formula to find the coordinates of R. In this case, m = 1, n = 5, x₁ = 4, y₁ = 8, x₂ = 11, and y₂ = 4. Substituting these values into the section formula, we get:

Calculating the x-coordinate of R

x = (1 * 11 + 5 * 4) / (1 + 5) x = (11 + 20) / 6 x = 31 / 6 x ≈ 5.17

Calculating the y-coordinate of R

y = (1 * 4 + 5 * 8) / (1 + 5) y = (4 + 40) / 6 y = 44 / 6 y ≈ 7.33

Therefore, the coordinates of point R are approximately (5.17, 7.33). This calculation demonstrates the practical application of the section formula in determining the exact location of a point dividing a line segment in a specified ratio. The step-by-step approach ensures accuracy and clarity in solving such problems.

To ensure a clear understanding of how we arrived at the coordinates of point R, let's break down the solution step-by-step. This detailed approach will help in grasping the application of the section formula and ensure no step is overlooked.

Step 1 Identifying the Given Values

First, we identify the given values from the problem statement:

  • Ratio of division: m:n = 1:5
  • Coordinates of point E: (x₁, y₁) = (4, 8)
  • Coordinates of point F: (x₂, y₂) = (11, 4)

Step 2 Applying the Section Formula for the x-coordinate

Using the section formula for the x-coordinate: x = (m * x₂ + n * x₁) / (m + n) Substitute the given values: x = (1 * 11 + 5 * 4) / (1 + 5) x = (11 + 20) / 6 x = 31 / 6 x ≈ 5.17

Step 3 Applying the Section Formula for the y-coordinate

Next, we apply the section formula for the y-coordinate: y = (m * y₂ + n * y₁) / (m + n) Substitute the given values: y = (1 * 4 + 5 * 8) / (1 + 5) y = (4 + 40) / 6 y = 44 / 6 y ≈ 7.33

Step 4 Combining the Coordinates

Finally, we combine the calculated x and y coordinates to find the coordinates of point R:

  • Coordinates of R: (5.17, 7.33)

This step-by-step solution illustrates the methodical approach to solving this type of problem. Each step is clearly defined, making it easy to follow and understand the process of using the section formula.

To ensure the accuracy of our solution, we can verify the result by checking if the calculated coordinates of point R indeed divide the line segment EF in the ratio 1:5. One way to do this is by calculating the distances ER and RF and confirming that their ratio is approximately 1:5.

Distance Formula

First, we recall the distance formula between two points (x₁, y₁) and (x₂, y₂):

Distance = √((x₂ - x₁)² + (y₂ - y₁)²)

Calculating Distance ER

Using the coordinates E(4, 8) and R(5.17, 7.33):

ER = √((5.17 - 4)² + (7.33 - 8)²) ER = √((1.17)² + (-0.67)²) ER = √(1.3689 + 0.4489) ER = √1.8178 ER ≈ 1.35

Calculating Distance RF

Using the coordinates R(5.17, 7.33) and F(11, 4):

RF = √((11 - 5.17)² + (4 - 7.33)²) RF = √((5.83)² + (-3.33)²) RF = √(33.9889 + 11.0889) RF = √45.0778 RF ≈ 6.71

Calculating the Ratio ER RF

Now, we calculate the ratio of ER to RF:

Ratio = ER / RF Ratio = 1.35 / 6.71 Ratio ≈ 0.20

Since 0.20 is approximately 1/5, our calculated coordinates for point R are consistent with the given ratio. This verification step confirms that our solution is accurate and reliable.

In conclusion, we have successfully determined the coordinates of point R that divides the line segment EF in the ratio 1:5. By applying the section formula, we found the coordinates of R to be approximately (5.17, 7.33). The detailed step-by-step solution and the subsequent verification provide a comprehensive understanding of the process. This problem highlights the importance of the section formula in coordinate geometry and its application in various mathematical and real-world scenarios. Mastering this concept is essential for anyone studying geometry and related fields. The ability to accurately find the coordinates of a dividing point is a valuable skill with numerous practical applications.

The final answer is (C) (5.17, 7.33)