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Chapter 40: Problem 5

These figures are similar. What is the length of \(z ?\) (A) 3 (B) 4 (C) 6 (D) 8

Short Answer

Expert verified
Since all the options give equal ratios, more information about side lengths a, b, and c or about the similarity ratio k is required to uniquely determine the value of z. In the current form, the given exercise is incomplete, and we cannot find the length of z without additional information.

Step by step solution

01

Set up the proportion

To set up the proportion, we can divide the side lengths of the smaller triangle by the corresponding side lengths of the larger triangle: \[\frac{a}{c}=\frac{b}{d}=\frac{z}{3z}\]
02

Solve for z

Looking at the proportion, we can see that we can use any of the equal ratios to solve for z. Since we have \(\frac{z}{3z}\), we can use this ratio to find z: \[\frac{z}{3z} = k\] Where k is the constant of proportionality. Now, we want to find the value of \(z\). To do so, we need to examine the four multiple choice options: 3, 4, 6, and 8. We can plug in each option for \(z\) and find the corresponding value of \(3z\).
03

Use the answer choices to find z

Option (A) z = 3: In this case, \(3z = 3 \times 3 = 9\). Option (B) z = 4: In this case, \(3z = 3 \times 4 = 12\). Option (C) z = 6: In this case, \(3z = 3 \times 6 = 18\). Option (D) z = 8: In this case, \(3z = 3 \times 8 = 24\).
04

Check which option gives equal ratios

Now, we want to check which option for \(z\) gives equal ratios when plugged back into the proportion. Option (A) \(\frac{3}{9} = \frac{1}{3}\) Option (B) \(\frac{4}{12} = \frac{1}{3}\) Option (C) \(\frac{6}{18} = \frac{1}{3}\) Option (D) \(\frac{8}{24} = \frac{1}{3}\) As all the options give equal ratios, we would require more information about side lengths a, b, and c or about the similarity ratio k to uniquely determine the value of z. In the current form, the given exercise is incomplete, and we cannot find the length of z without additional information.

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Key Concepts

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

Similar Figures
Similar figures are shapes that have the same form but different sizes. This means their corresponding angles are equal, and the ratios of the lengths of their corresponding sides are consistent. This concept is crucial because it allows us to understand and manipulate geometric shapes by using proportions.

To identify similar figures, you can look for the following characteristics:
  • All corresponding angles are equal.
  • The ratios between the lengths of corresponding sides are the same.
Knowing these properties can immensely simplify solving geometry problems, as it allows us to use these ratios to find missing side lengths or angles.
Proportions in Geometry
Proportions in geometry are essential for working with similar figures. A proportion is an equation that shows two ratios are equal. When figures are similar, you can set up a proportion that relates the corresponding sides of the figures.

For example, if you have two similar triangles, one smaller and one larger, you can create a ratio from each triangle and equate them. If the sides of the smaller triangle are represented as \( a, b, z \), and the larger as \( c, d, 3z \), you form proportions such as \( \frac{a}{c} = \frac{b}{d} = \frac{z}{3z} \).

These equations help solve for unknown values, ensuring you keep the sizes proportional, maintaining the aspect ratio of the shapes.
Solving for Variables in Math
To solve for variables in math accurately, especially in geometric problems, you must understand how to manipulate equations. Start with setting up a proportion based on the known and unknown values.

In the given exercise, where \( z \) is unknown, you equate the ratio with unknown variables to a constant. However, as shown, without additional information about the side lengths or the similarity ratio \( k \), finding the exact value of \( z \) is impossible.

Additionally, practicing these steps can enhance your problem-solving skills:
  • Substitute known values and simplify the equations.
  • Use logical reasoning to choose the most reasonable answer from available options.
  • Re-examine the problem if the results seem ambiguous or incomplete, as might happen in exercises lacking sufficient data.
Understanding this methodical approach to solving for variables will benefit you across various math and geometry problems.

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Most popular questions from this chapter

This shape is both a rectangle and a (A) parallelogram. (B) rhombus. (C) trapezoid. (D) square.

Use the formula to determine the measure of an interior angle in a regular hexagon. $$\frac{(n-2) \cdot 180^{\circ}}{n}$$ (a) \(30^{\circ}\) (B) \(60^{\circ}\) (C) \(110^{\circ}\) (D) \(120^{\circ}\)

Which volume formula should be used for a right triangular prism? A \(V=s^{3}\) (B) \(V=B h\) (C) \(V=/ w h\) (D) \(V=\frac{4}{3} \pi r^{3}\)

Which trigonometric equation is correct? (A) \(\sin 30^{\circ}=\frac{10}{x}\) (B) \(\cos 30^{\circ}=\frac{10}{x}\) (C) \(\tan 30^{\circ}=\frac{x}{10}\) (D) \(\sin 60^{\circ}=\frac{x}{10}\)
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