Question

Find the area triangle formed by the given three points. \((1,1),(2,3),\) and (3,3)

Short answer

The area of the triangle is 1 square unit.

Step-by-step solution

Identify the formula for the Area of a Triangle

The area of a triangle given its vertices \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) can be calculated using the determinant formula: \[ Area = \frac{1}{2} \, | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \]

Substitute the given points into the formula

Substitute the points \(1,1\), \(2,3\), and \(3,3\) into the formula to find the area: \[ Area = \frac{1}{2} \, | 1(3-3) + 2(3-1) + 3(1-3) | \]

Simplify the expression within the absolute value

Calculate the expression inside the absolute value: \[ 1(3-3) = 0, \quad 2(3-1) = 4, \quad 3(1-3) = -6 \]Adding them gives: \[ 0 + 4 - 6 = -2 \]

Calculate the absolute value and the area

The absolute value of \(-2\) is \(|-2| = 2\). Thus, the area is: \[ Area = \frac{1}{2} \times 2 = 1 \]

Key concepts

Area of a Triangle

The area of a triangle is a fundamental concept in geometry. It refers to the amount of space enclosed by the three sides of a triangle. In coordinate geometry, where the triangle is defined by the position of its three vertices, calculating the area becomes a straightforward task using a formula derived from the coordinates. You're essentially working with the vertices' coordinates to form a plane on which the area can be easily calculated.

The specific formula we use in this context is:

• \ Area = \frac{1}{2} \, | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) | \

This formula employs the coordinates \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) to find the area of the triangle they form.

Using the absolute value ensures that the area is always positive, as it should be a numerical representation of space. Calculating the area this way helps in determining the space no matter how the vertices are ordered; you're always assured of an accurate, positive result.

Determinant Formula

The determinant formula is a mathematical tool that you'll often encounter in linear algebra. It's especially handy here in finding the area of a triangle based on its vertices' coordinates.

Think of the determinant as a way to encase the calculations required to find the area based on where the vertices lie on a plane. By using this method, you're methodically adjusting the values based on each pair of coordinates.

• The core of the formula involves multiplying coordinates in a specific manner: \ x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \
• The absolute value ensures correctness in measurement, avoiding negative area results which don't have practical meaning.
• Finally, multiplying by \(\frac{1}{2}\) scales the value down to accurately reflect the triangle’s area in the coordinate plane.

This formula generalizes the area computation and makes it universally applicable not just for any triangular set of points, but for solving broader geometric problems as well.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a fascinating field that merges algebra and geometry by placing shapes within a coordinate system. This allows for a more rigorous understanding of spatial relationships. With triangles, understanding their shape and size through their vertices is particularly useful.

In coordinate geometry, you can plot points on a Cartesian plane, defined by \(x, y\) coordinates. This direct representation allows us to calculate properties like lengths, angles, and areas using algebraic formulas. For our purpose:

• We use the coordinates of points like \(1,1\), \(2,3\), and \(3,3\) to determine various parameters.
• The ability to use algebraic techniques to perform geometric operations simplifies complex calculations like area finding.
• This approach is elegant and efficient for tracing and understanding patterns or specific characteristics of a shape.

Straddling the line between pure mathematical thinking and practical application, coordinate geometry offers tools that are both powerful and intuitive, streamlining many steps otherwise complicated by abstract thinking alone.

Vertices of a Triangle

In geometry, especially when working with coordinate systems, vertices play a pivotal role in defining a shape. A triangle's vertices are its three distinct points which form its corners and dictate its size and shape.

For a triangle on a coordinate plane, vertices \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) are crucial:

• They determine the triangle's orientation and overall geometry.
• These points are used in the determinant formula to compute the precise area of the triangle.
• Knowing the exact locations of the vertices allows you to explore additional properties like perimeter and centroid placement.

Thus, simply knowing the vertices empowers you to uncover a multitude of geometric properties. By understanding precisely where your triangle sits in the coordinate plane, you can seamlessly transition between numeric calculations and visual representations, enriching your comprehension and capability to solve geometric problems.