Question

If the area of the quadrilatera! whose angular points taken in order are \((1,2),(-5,6),(7,-4)\) and \((h-2)\) be zero, find the value of \(h\) : (a) 2 (b) 3 (c) 5 (d) 6 :':

Short answer

Answer: h = 8/3 Explanation: Using the Shoelace Theorem, we found that the area of the quadrilateral can be represented by the equation 0 = (1/2)|6h - 16|. After solving this equation, we found that h = 8/3.

Step-by-step solution

Understand the Shoelace Theorem

The Shoelace Theorem states that the area of a simple polygon with vertices \((x_1,y_1),\) \((x_2,y_2),\) ...,\((x_n,y_n)\) is given by: \(Area = \frac{1}{2} \Big| \sum_{i=1}^{n-1} x_iy_{i+1} + x_ny_1 - \sum_{i=1}^{n-1} y_ix_{i+1} - y_nx_1 \Big|\) In our case, \(n=4\) as we have a quadrilateral.

Plug in the coordinates of the given vertices

Let our quadrilateral's vertices be \((x_1,y_1)=(1,2),\) \((x_2,y_2)=(-5,6),\) \((x_3,y_3)=(7,-4),\) and \((x_4,y_4)=(h,-2)\). Apply the Shoelace Theorem: \(Area = \frac{1}{2} \Big| (1)(6) + (-5)(-4) + (7)(-2) + (h)(2) - (2)(-5) - (6)(7) - (-4)(h) - (-2)(1) \Big|\)

Solve for the area

Simplify the equation above: \(Area = \frac{1}{2} \Big| 6 + 20 - 14 + 2h + 10 - 42 + 4h + 2 \Big|\) \(Area = \frac{1}{2} \Big| 6h - 16 \Big|\)

Set the area to zero and solve for \(h\)

The problem states that the area of the quadrilateral is zero. Therefore, we set the equation above equal to zero and solve for \(h\): \(0 = \frac{1}{2} \Big| 6h - 16 \Big|\) Note that \(6h - 16 = 0\) or \(6h - 16 = - (6h - 16)\), because we are looking for the absolute value. First, let's solve \(6h - 16 = 0\): \(6h = 16\) \(h = \frac{16}{6} = \frac{8}{3}\) Now, let's solve \(6h - 16 = -(6h - 16)\): \(6h - 16 = -6h + 16\) \(12h = 32\) \(h = \frac{32}{12} = \frac{8}{3}\) Both cases lead to the same value of \(h\).

Compare the solution with the given options

Our solution for \(h\) is \(\frac{8}{3}\), which is not available in the given options. Thus, there might be an error in the problem statement or it might be a trick question with no matching answer among the provided options.

Key concepts

Calculating the Area of a Quadrilateral

The calculation of the area of a quadrilateral can often be nontrivial, particularly when the shape is irregular and the sides are not parallel to the coordinate axes. Traditional methods such as breaking the shape down into triangles can become cumbersome. However, with the Shoelace Theorem, the task becomes systematic and efficient.

The Shoelace Theorem applies to any simple polygon in a two-dimensional coordinate plane. It is a convenient method to find the area when the vertices are known and can be applied to both irregular and regular quadrilaterals. The 'shoelace' name comes from the crisscross pattern of multiplication similar to the lacing of shoes, where you multiply across the coordinates as shown in the solution. This 'crisscross' method results in half the difference between the sum of the products of the coordinates taken in one order and the sum of the products taken in the opposite order. Regardless of the vertices' arrangement, the theorem ensures that the computed area is always positive, thanks to the absolute value.

The Shoelace Theorem is elegant because it does not require you to know the side lengths or angles of the quadrilateral. All you need are the vertice positions. It's essential to be accurate while applying this method, as misplacing coordinates can lead to incorrect calculations. Remember to always work carefully through the alternating sums and confirm each step when applying the theorem.

Coordinate Geometry: The Foundation of Polygon Calculations

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through the use of a coordinate system. This involves equations and algebra, making it easier to work with geometric concepts like points, lines, and polygons. In our quadrilateral area problem, knowing the coordinates of each vertex allows us to utilize algebraic methods to calculate geometric properties.

Each vertex of a polygon corresponds to a pair of numbers, which are the x (horizontal) and y (vertical) coordinates in the coordinate plane. This precise formulation of geometric space is what allows us to use methods such as the Shoelace Theorem to determine areas. In a way, coordinate geometry bridges the divide between algebra and geometry, giving us a powerful toolkit to address problems that might seem purely geometric at first glance.

In the practice of coordinate geometry, it is crucial to keep track of the signs of the coordinates, as they indicate the vertices' location relative to the origin – the central point (0, 0) of the plane. A complete understanding of the coordinate system is fundamental for students who wish to master the calculation of areas in geometry.

Polygon Area Calculation Using Coordinates

When calculating the area of polygons, it's often easier to deal with regular shapes, like squares and rectangles, where simple length times width calculations suffice. However, in the realm of coordinate geometry, polygons are not always neatly aligned with the axes, necessitating more universal methods of calculation like the Shoelace Theorem.

When using the Shoelace Theorem for polygon area calculation, it's important to list the vertices in a cyclic order – that is, going around the polygon as if you were drawing it without lifting your pen. The cyclic order can be either clockwise or counter-clockwise, as long as the sequence is continuous and ends where it started. This ensures that the summing and subtracting in the shoelace pattern captures the full geometry of the polygon.

Following these guidelines contributes to the stunningly direct and efficient process of the Shoelace Theorem, turning polygon area calculation into a sequence of simple algebraic operations. By making sure the coordinates are accurate and the arithmetic is performed correctly, this method can help students swiftly and accurately find the area of any polygon presented in a coordinate plane.