Problem 47 - Algebra 2

Question

ACT/SAT Triangle \(A B C\) has vertices with coordinates \(A(-4,2), B(-4,-3)\) and \(C(3,-2) .\) After a dilation, triangle \(A^{\prime} B^{\prime} C^{\prime}\) has coordinates \(A^{\prime}(-12,6)\) \(B^{\prime}(-12,-9),\) and \(C^{\prime}(9,-6) .\) How many times as great is the perimeter of \(\triangle A^{\prime} B^{\prime} C^{\prime}\) as that of \(\triangle A B C ?\) A 3 B 6 C 12 D \(\frac{1}{3}\)

Step-by-Step Solution

Answer

Option A: 3

Step 1: Find the scale factor of dilation

To find the scale factor of dilation, we compare the coordinates of corresponding points before and after dilation. For example, compare point \(A\) with \(A'\):\(-12/(-4) = 3\) and \(6/2 = 3\). This shows the dilation scale factor is 3.

Step 2: Calculate the original perimeter of \(\triangle ABC\)

Find the length of each side of \(\triangle ABC\) using the distance formula. For side \(AB\): \[ \sqrt{(-4 - (-4))^2 + (2 - (-3))^2} = \sqrt{0 + 25} = 5 \] For side \(BC\): \[ \sqrt{(3 - (-4))^2 + (-2 - (-3))^2} = \sqrt{49 + 1} = \sqrt{50} \] For side \(CA\): \[ \sqrt{(3 - (-4))^2 + (-2 - 2)^2} = \sqrt{49 + 16} = \sqrt{65} \] The perimeter of \(\triangle ABC\) is \[ 5 + \sqrt{50} + \sqrt{65} \]

Step 3: Calculate the new perimeter of \(\triangle A'B'C'\)

Using the same method, find the lengths of sides of \(\triangle A'B'C'\):For side \(A'B'\): \[ \sqrt{(-12 - (-12))^2 + (6 - (-9))^2} = \sqrt{0 + 225} = 15 \] For side \(B'C'\): \[ \sqrt{(9 - (-12))^2 + (-6 - (-9))^2} = \sqrt{441 + 9} = \sqrt{450} \] For side \(C'A'\): \[ \sqrt{(9 - (-12))^2 + (-6 - 6)^2} = \sqrt{441 + 144} = \sqrt{585} \] The perimeter of \(\triangle A'B'C'\) is:\[ 15 + \sqrt{450} + \sqrt{585} \]

Step 4: Compare the perimeters

Since the dilation scale factor is 3, the perimeter of \(\triangle A'B'C'\) is exactly 3 times that of \(\triangle ABC\). The perimeters are related by the scale factor of the dilation. Thus, the ratio of the perimeter of \(\triangle A'B'C'\) to \(\triangle ABC\) is 3.

Key Concepts

Distance Formula

The distance formula is a handy tool in coordinate geometry. It helps us find the distance between two points in a plane. When dealing with triangles like \(\triangle ABC\), it's essential to find side lengths to calculate the perimeter.
The formula is: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• First, identify the coordinates of the two points.
• Subtract the x-coordinates and the y-coordinates separately.
• Square each difference, sum them, and then take the square root.

Using this formula, you can find the lengths of sides \(AB\), \(BC\), and \(CA\) in \(\triangle ABC\). For instance, for side \(AB\), using \(A(-4,2)\) and \(B(-4,-3)\), the distance is:\[ \sqrt{(-4 - (-4))^2 + (2 - (-3))^2} = \sqrt{0 + 25} = 5 \]

Perimeter

The perimeter of a triangle is the total length around the triangle. It is simply the sum of its three side lengths.
For \(\triangle ABC\), we calculate the perimeter by adding the lengths of \(AB\), \(BC\), and \(CA\) obtained using the distance formula:

• \(AB = 5\),
• \(BC = \sqrt{50}\),
• \(CA = \sqrt{65}\).

To find the perimeter:\[ 5 + \sqrt{50} + \sqrt{65} \]
This representation provides an exact value that avoids approximations until necessary, preserving precision. Understanding this helps when comparing perimeters of similar figures after transformations like dilation.

Scale Factor

The scale factor in dilation determines how much each dimension of a figure is enlarged or reduced. It measures the degree of scaling applied to a geometric shape.
To find the scale factor in coordinate geometry, compare corresponding coordinates of two figures.
For triangle \(\triangle A^{\prime} B^{\prime} C^{\prime}\) compared to \(\triangle ABC\), observe a point like \(A\) and \(A^{\prime}\):

• The x-coordinate comparison: \(-12/(-4) = 3\)
• The y-coordinate comparison: \(6/2 = 3\)

Both comparisons give a consistent scale factor of 3. This means every side length in \(\triangle A^{\prime} B^{\prime} C^{\prime}\) is three times as long as its corresponding side in \(\triangle ABC\). This scaling factor applies uniformly across the triangle, affecting both the sides and the perimeter.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate system to study the properties of geometric shapes. With coordinates, you can easily apply formulas like the distance formula and transformations such as dilation. This branch of mathematics allows us to solve problems related to points, lines, and triangles on a plane.

In solving the problem with \(\triangle ABC\), coordinate geometry enabled the accurate calculation of distances between points. Furthermore, it provided a method to determine the triangle perimeter before and after dilation using the scale factor.

Understanding the basics of coordinate geometry can simplify many problems related to shape transformations, measurements, and analysis in a two-dimensional space. This approach is critical for exams like the SAT/ACT, where precision and methodical application of mathematical tools are key.