Finding Coordinates Dividing A Line Segment In A Given Ratio

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Introduction

In geometry, understanding how to divide a line segment in a given ratio is a fundamental concept. This often involves finding the coordinates of a point that partitions the segment into specific proportions. This article delves into the method for determining the coordinates of such a point. We will use a specific example to illustrate the process, focusing on dividing the line segment $\\overlineEF}$\ in the ratio 15. This problem is a classic example of applying the section formula, a crucial tool in coordinate geometry. By working through this example, you will gain a solid understanding of how to apply the section formula and solve similar problems. Mastering this concept is essential for further studies in geometry and related fields. Let's explore the step-by-step approach to finding the coordinates of point R, which divides the line segment $\\overline{EF$\ in the given ratio.

Understanding the Section Formula

Before diving into the problem, it's crucial to understand the section formula. This formula helps us find the coordinates of a point that divides a line segment in a given ratio internally. Suppose we have a line segment joining two points, A($x_1$, $y_1$) and B($x_2$, $y_2$), and a point P($x$, $y$) that divides the segment AB in the ratio m:n. The coordinates of point P can be calculated using the following formulas:

x=fracmx2+nx1m+nx = \\frac{mx_2 + nx_1}{m + n}

y=fracmy2+ny1m+ny = \\frac{my_2 + ny_1}{m + n}

These formulas are derived from the principles of similar triangles and provide a straightforward method for finding the coordinates of the dividing point. In the context of our problem, we will apply these formulas to find the coordinates of point R, which divides the line segment $\\overline{EF}$\ in the ratio 1:5. Understanding the derivation and application of the section formula is paramount to solving problems related to dividing line segments in coordinate geometry. This formula is not only applicable in academic settings but also has practical uses in fields like engineering and computer graphics, where precise positioning and division of spaces are necessary.

Problem Statement: Dividing Line Segment EF

The problem states that point R divides the line segment $\\overline{EF}$\ in the ratio 1:5. The coordinates of point E are given as (4, 8), and the coordinates of point F are (11, 4). Our goal is to find the coordinates of point R, accurate to two decimal places. This means we need to apply the section formula correctly, substitute the given values, and perform the necessary calculations to obtain the x and y coordinates of point R. The precision requirement of two decimal places emphasizes the importance of accurate calculations and rounding off the final answers appropriately. This type of problem is commonly encountered in coordinate geometry and tests the understanding of section formula application. Solving it requires a clear understanding of the formula, correct substitution of values, and careful arithmetic calculations. Let's proceed step by step to find the coordinates of point R.

Step-by-Step Solution

  1. Identify the Given Values: We are given the coordinates of E as (4, 8) and the coordinates of F as (11, 4). The ratio in which R divides $\\overline{EF}$\ is 1:5. This means that m = 1 and n = 5.

  2. Apply the Section Formula: Using the section formula, we can find the x-coordinate ($x_R$) and the y-coordinate ($y_R$) of point R:

    xR=fracmxF+nxEm+n=frac1cdot11+5cdot41+5x_R = \\frac{m x_F + n x_E}{m + n} = \\frac{1 \\cdot 11 + 5 \\cdot 4}{1 + 5}

    yR=fracmyF+nyEm+n=frac1cdot4+5cdot81+5y_R = \\frac{m y_F + n y_E}{m + n} = \\frac{1 \\cdot 4 + 5 \\cdot 8}{1 + 5}

  3. Substitute the Values: Now, substitute the given values into the formulas:

    xR=frac1cdot11+5cdot41+5=frac11+206=frac316x_R = \\frac{1 \\cdot 11 + 5 \\cdot 4}{1 + 5} = \\frac{11 + 20}{6} = \\frac{31}{6}

    yR=frac1cdot4+5cdot81+5=frac4+406=frac446y_R = \\frac{1 \\cdot 4 + 5 \\cdot 8}{1 + 5} = \\frac{4 + 40}{6} = \\frac{44}{6}

  4. Calculate the Coordinates: Perform the calculations to find the coordinates of R:

    xR=frac316approx5.17x_R = \\frac{31}{6} \\approx 5.17

    yR=frac446approx7.33y_R = \\frac{44}{6} \\approx 7.33

  5. Round to Two Decimal Places: Round the coordinates to two decimal places as required:

    xRapprox5.17x_R \\approx 5.17

    yRapprox7.33y_R \\approx 7.33

Therefore, the coordinates of point R are approximately (5.17, 7.33).

Detailed Calculation of X-Coordinate

To find the x-coordinate of point R, we apply the section formula specifically for the x-coordinate. The formula is given by: $x_R = \fracm x_F + n x_E}{m + n}$. In our problem, the ratio mn is 1:5, so m = 1 and n = 5. The x-coordinate of point E ($x_E$) is 4, and the x-coordinate of point F ($x_F$) is 11. Substituting these values into the formula, we get: $x_R = \frac{1 \cdot 11 + 5 \cdot 4{1 + 5}$. This simplifies to $x_R = \frac{11 + 20}{6}$. Further simplification yields $x_R = \frac{31}{6}$. To express this as a decimal, we divide 31 by 6, which gives us approximately 5.16666... Since we need to round the answer to two decimal places, we look at the third decimal place. In this case, it is 6, which is greater than or equal to 5, so we round up the second decimal place. Therefore, the x-coordinate of point R, rounded to two decimal places, is approximately 5.17. This step-by-step calculation ensures accuracy and clarity in finding the x-coordinate of the point that divides the line segment in the given ratio.

Detailed Calculation of Y-Coordinate

Next, we calculate the y-coordinate of point R using the section formula for the y-coordinate. The formula is given by: $y_R = \fracm y_F + n y_E}{m + n}$. Again, the ratio mn is 1:5, so m = 1 and n = 5. The y-coordinate of point E ($y_E$) is 8, and the y-coordinate of point F ($y_F$) is 4. Substituting these values into the formula, we get: $y_R = \frac{1 \cdot 4 + 5 \cdot 8{1 + 5}$. This simplifies to $y_R = \frac{4 + 40}{6}$. Further simplification yields $y_R = \frac{44}{6}$. To express this as a decimal, we divide 44 by 6, which gives us approximately 7.33333... As with the x-coordinate, we need to round the answer to two decimal places. The third decimal place is 3, which is less than 5, so we round down, keeping the second decimal place as it is. Therefore, the y-coordinate of point R, rounded to two decimal places, is approximately 7.33. This detailed calculation ensures we accurately determine the y-coordinate of point R, completing the process of finding the coordinates of the dividing point.

Final Answer and Conclusion

In conclusion, by applying the section formula, we have determined the coordinates of point R, which divides the line segment $\\overline{EF}$\ in the ratio 1:5. The coordinates of point E are (4, 8), and the coordinates of point F are (11, 4). After performing the calculations and rounding to two decimal places, we found that the coordinates of point R are approximately (5.17, 7.33). This result provides a precise location of point R on the coordinate plane, fulfilling the requirements of the problem. The section formula is a powerful tool in coordinate geometry, allowing us to find the coordinates of a point that divides a line segment in any given ratio. This concept is widely applicable in various fields, including geometry, calculus, and computer graphics. Understanding and mastering the section formula is crucial for solving a wide range of problems related to line segments and coordinate geometry. This exercise demonstrates a practical application of the formula and reinforces the importance of accuracy in calculations and rounding off to the required decimal places.