Question

Two poles of different lengths are placed against a wall at identical angles, forming two similar triangles. Select two numbers from those below that give possible values for the length of the shorter pole and the height up the wall that it reaches. (For this practice test, write the numbers in the boxes above.) 4 5 6 8 11

Short answer

The two possible values for the length of the shorter pole and the height up the wall that it reaches are 4 and 8.

Step-by-step solution

List given numbers and find their possible pairs.

We are given the numbers 4, 5, 6, 8, and 11. Let's list the possible pairs of numbers that these can form: (4, 5), (4, 6), (4, 8), (4, 11), (5, 6), (5, 8), (5, 11), (6, 8), (6, 11), and (8, 11).

Identify the pairs that have the same ratio.

We must find pairs of numbers from the list in step 1 that maintain the same ratio, as this will show that the triangles formed are similar. We can calculate the ratio of each pair in the list and compare them: (4, 5): \(\frac{4}{5} = 0.8\) (4, 6): \(\frac{4}{6} = 0.666...\) (4, 8): \(\frac{4}{8} = 0.5\) (4, 11): \(\frac{4}{11} = 0.363...\) (5, 6): \(\frac{5}{6} = 0.833...\) (5, 8): \(\frac{5}{8} = 0.625\) (5, 11): \(\frac{5}{11} = 0.454...\) (6, 8): \(\frac{6}{8} = 0.75\) (6, 11): \(\frac{6}{11} = 0.545...\) (8, 11): \(\frac{8}{11} = 0.727...\) There are no pairs of numbers with the same ratio in this list.

Consider the larger pole's length as an integer multiple of the smaller pole and compare with the given numbers.

Since both triangles are similar, the lengths of the shorter and longer poles must have an integer ratio. Let's consider the length of the larger pole as an integer multiple of the smaller pole and compare this with the given numbers. Let the length of the larger pole be twice the smaller pole. The length of the larger pole can be two times either of the given numbers. Let's first consider the smaller pole to be 4: If the larger pole is twice the length of 4, then it would be 8. This pair (4, 8) is also available in the given numbers. In this case, since the poles have a ratio of 1:2, the height that the poles reach up the wall must also have a ratio of 1:2. Since \(4 < 8\), we can consider 4 as the length of the smaller pole and 8 as the height it reaches up the wall. We now have our answer: The two possible values for the length of the shorter pole and the height up the wall that it reaches are 4 and 8.

Key concepts

Similar Triangles

Understanding similar triangles is essential when studying geometry, especially in problems that involve comparing shapes or proportions. Similar triangles are those that have the same shape but may differ in size. This means all corresponding angles between two similar triangles are equal and their corresponding sides are in proportion.

For instance, if two poles are leaning against a wall at equal angles, they create triangles with the wall and the ground that are similar. The length of the poles and the height they reach on the wall will be in the same ratio. This is exactly what makes them 'similar'—their corresponding angles are congruent and the lengths of their corresponding sides are proportional.

Ratios

Ratios are a fundamental concept in mathematics that express the relationship between two quantities. It tells us how many times one value contains another. Ratios can be presented in different forms such as fractions, decimals, or with a colon, like 3:1.

When working with similar triangles, ratios help determine the proportionality of the corresponding sides. In the context of the exercise, the ratios between the lengths of the shorter pole and the height up the wall it reaches are sought. By comparing the ratios of different pairs, one can decipher which sets of numbers could represent the real-world lengths and heights of the poles and the wall according to the principles of similar triangles.

Mathematical Reasoning

Mathematical reasoning involves the process of thinking logically and methodically about number and quantitative relationships. In problems involving similar triangles and ratios, it helps to reason through a step-by-step process, assessing which number pairs exhibit the required proportionality.

The solution to the exercise demonstrated a systematic approach, first listing all possible pairs of the given numbers, then calculating their ratios, and finally considering the real-world context that the ratio between the lengths of the poles must be an integer. Mathematical reasoning permeates every step — from understanding what similar triangles imply about ratios to applying the concept to a practical example involving poles and a wall.