Problem 45 - Algebra 2

Question

For Exercises 44 and \(45,\) use the following information. The vertices of \(\triangle A B C\) are \(A(-2,1), B(1,2)\) and \(C(2,-3) .\) The triangle is dilated so that its perimeter is 2\(\frac{1}{2}\) times the original perimeter. Find the coordinates of \(\triangle A^{\prime} B^{\prime} C^{\prime} .\) Then graph \(\triangle A B C\) and \(\triangle A^{\prime} B^{\prime} C^{\prime}\)

Step-by-Step Solution

Answer

Coordinates of \( \triangle A'B'C' \) are \((-5, 2.5), (2.5, 5), (5, -7.5)\).

Step 1: Calculate the Original Side Lengths

To find the perimeter of \( \triangle ABC \), we first calculate the lengths of the sides. Use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for each pair of coordinates. Calculate \( AB \):\[ d_{AB} = \sqrt{(1 - (-2))^2 + (2 - 1)^2} = \sqrt{3^2 + 1^2} = \sqrt{10} \]Calculate \( BC \):\[ d_{BC} = \sqrt{(2 - 1)^2 + (-3 - 2)^2} = \sqrt{1^2 + (-5)^2} = \sqrt{26} \]Calculate \( CA \):\[ d_{CA} = \sqrt{(2 - (-2))^2 + (-3 - 1)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{32} \]

Step 2: Calculate Original Perimeter

Add the side lengths to find the perimeter of \( \triangle ABC \):\[ P = \sqrt{10} + \sqrt{26} + \sqrt{32} \]

Step 3: Calculate New Perimeter

According to the problem, the new perimeter is 2\(\frac{1}{2}\) times the original perimeter. New perimeter = \( 2.5 \times P \)

Step 4: Calculate Scale Factor

Because the scale factor applied in dilation affects the triangle uniformly, let's denote the scale factor by \( k \). Since the perimeter increases by a factor of 2.5, and perimeter scales linearly with side lengths, the scale factor \( k \) is also 2.5.

Step 5: Apply Dilation to Vertices

Apply the dilation to each vertex using the origin \((0,0)\) as the center. Each coordinate of a vertex is multiplied by the scale factor:For \( A' \): \[ A' = (2.5 \times -2, 2.5 \times 1) = (-5, 2.5) \]For \( B' \):\[ B' = (2.5 \times 1, 2.5 \times 2) = (2.5, 5) \]For \( C' \):\[ C' = (2.5 \times 2, 2.5 \times -3) = (5, -7.5) \]

Step 6: Plot Coordinates on Graph

Plot the original vertices \((-2,1), (1,2), (2,-3)\) of \( \triangle ABC \) and the dilated vertices \((-5, 2.5), (2.5, 5), (5, -7.5)\) of \( \triangle A'B'C' \) on a graph. This visualization will help confirm the accuracy of the dilation process.

Key Concepts

Triangle Vertices

In any geometric shape, vertices are crucial as they define the corners of the figure. For a triangle, which is a polygon with three sides, there are exactly three vertices. In our exercise, we have triangle \( \triangle ABC \) with the vertices given by the coordinates \( A(-2,1) \), \( B(1,2) \), and \( C(2,-3) \). Each vertex is represented by a pair of numbers within the coordinate plane, indicating where that point lies on the plane.
Understanding vertices involves grasping how to plot these points accurately on a graph. Start by finding each vertex's position using the x-coordinate, which tells you how far left or right to go, and the y-coordinate, which tells you how far up or down to go. Plotting these will give you a clear visual of the triangle's shape.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves representing geometric figures in a coordinate plane using algebraic equations. The coordinate plane is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis).
By knowing the coordinates of points like \( A(-2,1) \), \( B(1,2) \), and \( C(2,-3) \), you can calculate distances and angles, and apply transformations, such as dilations. Dilation in geometry involves resizing a shape, and is done by multiplying the coordinates of the shape's vertices by a scale factor while keeping their relative structure intact.
In this exercise, we used a scale factor of 2.5 to dilate \( \triangle ABC \), resulting in new vertices \( A'(-5, 2.5) \), \( B'(2.5, 5) \), and \( C'(5, -7.5) \). This process highlights how coordinate geometry allows us to manipulate geometric figures systematically.

Distance Formula

The distance formula is a fundamental tool in coordinate geometry that helps find the distance between two points in the coordinate plane. Given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the distance \( d \) between them can be calculated using:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

In our exercise, this formula was used to calculate the lengths of the sides of \( \triangle ABC \):

• For \( AB \), points \((x_1, y_1) = (-2, 1)\) and \((x_2, y_2) = (1, 2)\). Distance \( d_{AB} = \sqrt{10} \).
• For \( BC \), points \((1, 2)\) and \((2, -3)\). Distance \( d_{BC} = \sqrt{26} \).
• For \( CA \), points \((2, -3)\) and \((-2, 1)\). Distance \( d_{CA} = \sqrt{32} \).

These calculations are crucial for determining the perimeter and understanding the impact of dilation.

Perimeter Calculation

Perimeter is the total distance around a polygon and is calculated by summing the lengths of all its sides. In the case of \( \triangle ABC \), the perimeter \( P \) is calculated as follows:

• \( P = \sqrt{10} + \sqrt{26} + \sqrt{32} \)

Perimeter calculation helps in understanding how dilation affects geometric figures. In this problem, the triangle's perimeter was dilated by a factor of 2.5, meaning the new perimeter became 2.5 times the original. This is calculated by multiplying the original perimeter \( P \) by 2.5.
With the new perimeter set, we deduced that the scale factor for dilation (also 2.5) was uniformly applied to each vertex's coordinates to obtain the vertices of the dilated triangle \( \triangle A'B'C' \). This process underscores the relationship between side lengths, perimeter, and transformation in geometry.