Question

The co-ordinates of \(A, B, C\) are \((6,3),(-3,5),(4,-2)\) respectively and \(P\) is a point \((x, y)\), then find the value of \(\frac{\Delta P B C}{\Delta A B C}:\) (a) \(\frac{x+y-2}{7}\) (b) \(\frac{x-y}{4}\) \(\therefore 6\) (c) \(\frac{x+y-3}{5}\) (d) none of these

Short answer

Q: Calculate the ratio of the area of triangle \(PBC\) to the area of triangle \(ABC\), given the coordinates of points \(A (6, 3)\), \(B (-3, 5)\), \(C (4, -2)\), and \(P (x, y)\). A: The correct ratio is given by \(\frac{|3y - 3x + 30|}{49}\) which is not equal to any of the given options. So the correct answer is (d) none of these.

Step-by-step solution

Calculate the area of triangle ABC

To calculate the area of triangle ABC, we can use the Shoelace Formula, which states that the area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is: \(\Delta = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + x_3 y_1) - (y_1 x_2 + y_2 x_3 + y_3 x_1)|\) Substitute the vertices coordinates (6,3), (-3,5), and (4,-2) into the formula: \(\Delta ABC = \frac{1}{2} |(6 \cdot 5 + (-3) \cdot (-2) + 4 \cdot 3) - (3 \cdot (-3) + 5 \cdot 4 + (-2) \cdot 6)|\) = \(\frac{1}{2}|(30 + 6 + 12) - (-9 + 20 - 12)|\) = \(\frac{1}{2}|(48 - (-1))|\) = \(\frac{1}{2} \cdot 49\) = \(24.5\)

Calculate the area of triangle PBC

To calculate the area of triangle PBC, we can use the Shoelace Formula again, substituting the vertices coordinates of \(P(x, y)\), \(B(-3,5)\), and \(C(4,-2)\): \(\Delta PBC = \frac{1}{2} |(-3y + 5x - 2x) - (5(-3) - 2(-3) + x(5 - (-2)))|\) = \(\frac{1}{2} |(3y + 2x + 9) + 15 + 6 - 5x + 2x)|\) = \(\frac{1}{2} |(3y - 3x + 30)|\)

Find the ratio

Now we can find the ratio of the area of triangle PBC to the area of triangle ABC: \(\frac{\Delta PBC}{\Delta ABC} = \frac{\frac{1}{2} |(3y - 3x + 30)|}{24.5}\) Cancel out the \(\frac{1}{2}\): \(\frac{\Delta PBC}{\Delta ABC} = \frac{|3y - 3x + 30|}{49}\) Now compare this result with the given options: (a) \(\frac{x+y-2}{7}\) (b) \(\frac{x-y}{4}\) (c) \(\frac{x+y-3}{5}\) (d) none of these None of the given options match the result we found above, so the correct answer is: (d) none of these

Key concepts

Triangle Area Calculation

Calculating the area of a triangle in coordinate geometry can be done using various methods, but one of the most straightforward is the Coordinate Geometry method, specifically the Shoelace Formula. This formula helps us determine the area when the vertices of the triangle are given in coordinate form. It operates on the principle that given three vertices, \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), we can find the area by: \[\Delta = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + x_3 y_1) - (y_1 x_2 + y_2 x_3 + y_3 x_1)|\] The basic idea here is that when calculating the determinants formed by taking products of the coordinates, we get individual triangles whose signed areas sum to the area of the entire triangle.
This method is not only effective but also conveniently deals with positive and negative numbers by using absolute values, ensuring we get a non-negative area.
You'll often find this method discussed in context with the term "shoelace" due to how you might visualize the process as lacing across and back through each coordinate.

Shoelace Formula

The Shoelace Formula, also known as Gauss's area formula, is a nifty mathematical tool used to determine the area of a polygon when its vertices are known.This formula is immensely helpful in scenarios involving triangles, particularly in coordinate geometry.
In the formula, listed are coordinates of the vertices, and the algebraic operations resemble crisscrossing shoelaces. That's the origine of its unique name.

• Write down the coordinates of all vertices, repeating the first vertex at the end of the list.
• Perform summation for the products of diagonally connected vertices from left to right (\(x_i y_{i+1}\)).
• Subtract the summation of products from right to left (\(y_i x_{i+1}\)).
• Take the absolute difference, divide by 2, and you obtain the geometric area of the polygon.

For triangles, this process is particularly streamlined since they only have three vertices.
Using such a formula is advantageous because it reduces errors that may result from manual calculations of small segments of the triangle. Its reliability and straightforward nature make it a popular choice for computations in both academic and practical applications.

Ratio of Areas

Understanding the concept of the ratio of areas can be particularly useful when comparing parts of similar geometric shapes.When tasked with finding the ratio of the areas of two triangles located on the same plane or coordinate system, the principle remains the division of their respective areas.
Mathematically, this can be written as: \[\text{Ratio} = \frac{\Delta_1}{\Delta_2}\] Here, \(\Delta_1\) and \(\Delta_2\) denote the areas of the two triangles.

• This ratio can provide insight into relative sizes, angles, or positions on a plane.
• The simplifying principle involves the factual division, making it easy to numerically understand the geometric relationship between shapes.
• Ratios highlight symmetrical relationships and provide a dimensionless measure that is simply a comparison of the two values, rather than providing the actual sizes.

For example, in our problem, seeing that none of the provided options matched the computed ratio,we verified that the triangles' relationship based on their dimensions did not fit predetermined expectations, leading to the conclusion: "none of these."
This emphasizes the importance of precise calculations and proper understanding to draw accurate mathematical conclusions.