/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Determine the area of \(\triangl... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 35

Determine the area of \(\triangle A B C\) if \(A=(2,1), B=(5,3)\) and \(C\) is the reflection of \(B\) across the \(x\) axis.

Short Answer

Expert verified
The area of \(\triangle ABC\) is 9 square units.

Step by step solution

01

Determine Point C

The point C is the reflection of point B across the x-axis. To reflect a point across the x-axis, we change the sign of its y-coordinate. Since B is at (5, 3), the reflected point C will be at (5, -3).
02

Use the Formula for Area of a Triangle

The formula for the area of a triangle formed by points \(A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\) is \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \].
03

Insert the Coordinates into the Formula

Using the coordinates A(2, 1), B(5, 3), and C(5, -3), we insert them into the formula. This gives us \[ \text{Area} = \frac{1}{2} \left| 2(3 + 3) + 5(-3 - 1) + 5(1 - 3) \right| \].
04

Calculate the Expression Inside Absolute Value

Calculate the expression: \((2 \times 6) + (5 \times -4) + (5 \times -2) = 12 - 20 - 10 = -18\).
05

Calculate the Absolute Value and Area

Take the absolute value of the expression: \| -18 | = 18\. Then, compute the area: \( \frac{1}{2} \times 18 = 9 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Area
To find the area of a triangle, especially when working with points on a coordinate plane, a specific formula comes in handy. This is not the standard base times height over two formula but a more flexible one that utilizes coordinates directly. Imagine three points: \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\).
\[\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]
This formula is a powerful tool in coordinate geometry. It allows you to determine the area by plugging in the x and y values directly, making it extremely convenient.
  • Calculate the differences in y-coordinates for each pair of points.
  • Multiply each x-coordinate by the respective y-pair difference.
  • Add these calculated values together.
  • Take the absolute value to ensure the area is positive.
  • Divide the result by 2 to get the final area.
This approach simplifies the process and eliminates the need for drawing or measuring actual base and height.
Coordinate Geometry
Coordinate Geometry provides a way to describe geometric figures using an algebraic approach. It establishes a link between algebra and geometry through the Cartesian coordinate system, where every point is defined by an \(x\) (horizontal position) and a \(y\) (vertical position).
  • Each point on the coordinate plane is expressed as \((x, y)\).
  • Vertical and horizontal distances can be calculated directly from these coordinates.
  • Equations of lines and shapes can be easily formed using the coordinates.
In the context of triangles formed by coordinates, you can place points anywhere in the plane and use the coordinate formula to determine properties like area or even check for collinearity. This makes coordinate geometry an essential tool for solving a wide range of geometric problems without needing a physical diagram or visual representation.
Reflection across Axes
Reflection across an axis is a fundamental transformation in geometry, particularly coordinate geometry. It involves flipping a point, line, or shape over either the x-axis or the y-axis. When reflecting a point such as \((x, y)\):
  • Over the \(x\)-axis: change \(y\) to \(-y\). Thus, \((x, y)\) becomes \((x, -y)\).
  • Over the \(y\)-axis: change \(x\) to \(-x\). Thus, \((x, y)\) becomes \((-x, y)\).
This concept is particularly useful when determining positions relative to axes. By reflecting, you can quickly determine points that are symmetric along the axis.
The reflection concept applies to any point or object on the coordinate plane, and it helps in visualizing symmetrical properties and simplifying geometric problems that involve symmetry.

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Most popular questions from this chapter

Describe the steps of the procedure that enables us to find the distance from a point \(P(a, b)\) to the line \(A x+B y=C\). (GRAPH CAN'T COPY)

Points \(A\) and \(B\) have symmetry with respect to the vertical line where \(x=2 .\) Find the coordinates of \(A\) if \(B\) is the point: a) \((5,1)\) b) \((0,5)\) c) \((-6, a)\) d) \((b, c)\)

Find the equation of the line described. Leave the solution in the form \(A x+B y=C\). The line contains \((b, c)\) and has slope \(d\).

Find the equation of the line described. Leave the solution in the form \(A x+B y=C\). The line has intercepts \(a=2\) and \(b=-2\).

Apply the Midpoint Formula. A circle has its center at the point \((-2,3) .\) If one endpoint of a diameter is at \((3,-5),\) find the other endpoint of the diameter.
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