Similar Triangles Determining Side Lengths Of Non Congruent Triangles

by Scholario Team 70 views

In geometry, understanding the properties of similar triangles is crucial. Similar triangles have the same shape but can differ in size. This means their corresponding angles are equal, and their corresponding sides are in proportion. When we discuss triangles, key concepts such as side lengths, congruence, and similarity come into play. This article will delve into how to determine the side lengths of a triangle that is similar but not congruent to a given triangle. We will explore the principles of similarity, proportionality, and how to apply these concepts to solve geometric problems.

Defining Similarity and Congruence

Before diving into the problem, it’s essential to clarify the terms similarity and congruence. Two triangles are said to be similar if they have the same shape, meaning their corresponding angles are equal, and their corresponding sides are in proportion. The ratio of the lengths of corresponding sides is called the scale factor. On the other hand, two triangles are congruent if they are exactly the same – same shape and same size. Congruent triangles have equal corresponding sides and angles. To put it simply, similar triangles are scaled versions of each other, while congruent triangles are identical.

Analyzing Triangle ABC

The vertices of extit{triangle ABC} are given as A(1, -2), B(1, 1), and C(5, -2). To determine the side lengths of this triangle, we use the distance formula, which is derived from the Pythagorean theorem. The distance d between two points extit{(x1, y1)} and extit{(x2, y2)} in a coordinate plane is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Let’s calculate the lengths of the sides of extit{triangle ABC}:

  1. Length of AB: The coordinates of A and B are (1, -2) and (1, 1), respectively.

    AB = √((1 - 1)^2 + (1 - (-2))^2) = √(0 + 9) = √9 = 3 units

  2. Length of BC: The coordinates of B and C are (1, 1) and (5, -2), respectively.

    BC = √((5 - 1)^2 + (-2 - 1)^2) = √(16 + 9) = √25 = 5 units

  3. Length of AC: The coordinates of A and C are (1, -2) and (5, -2), respectively.

    AC = √((5 - 1)^2 + (-2 - (-2))^2) = √(16 + 0) = √16 = 4 units

Thus, the side lengths of extit{triangle ABC} are 3, 4, and 5 units. This is a classic Pythagorean triple, indicating that extit{triangle ABC} is a right-angled triangle.

Identifying Similar Triangles

A triangle that is similar to extit{triangle ABC} will have sides in the same ratio as 3:4:5. However, it should not be congruent, meaning its sides should not be exactly 3, 4, and 5 units. The sides must be a multiple of this ratio but not the same set of numbers. To find such a triangle, we look for a set of side lengths that are in the ratio 3:4:5 but scaled by a factor greater than 1.

Analyzing the Options

Now let's consider the given options and determine which could be the side lengths of a triangle that is similar but not congruent to extit{triangle ABC}:

Option A: 3, 4, and 5 units

These are the same side lengths as extit{triangle ABC}. Therefore, this triangle would be congruent to extit{triangle ABC}, not just similar. Hence, Option A is incorrect.

Option B: 9, 12, and 15 units

To check if these side lengths are in the same ratio as 3:4:5, we can divide each number by the greatest common divisor, which is 3:

9 / 3 = 3

12 / 3 = 4

15 / 3 = 5

Since the ratio is 3:4:5, this triangle is similar to extit{triangle ABC}. However, the side lengths are a multiple of the original triangle's side lengths (3 times larger), so it is not congruent. Therefore, Option B is a potential answer.

Option C: 6, 4, and 5 units

To determine if these side lengths form a triangle similar to extit{triangle ABC}, we need to check if the ratios of the corresponding sides are equal. The sides of extit{triangle ABC} are 3, 4, and 5. If we try to match the smallest side of the new triangle (4) with the smallest side of extit{triangle ABC} (3), and so on, we will not find a consistent ratio. Thus, this triangle is not similar to extit{triangle ABC}, and Option C is incorrect.

Option D: 9, 12, and 5 units

In this set, the sides are 9, 12, and 5. If we check the ratios, we can quickly see that they do not match the 3:4:5 ratio of extit{triangle ABC}. Therefore, this triangle is not similar to extit{triangle ABC}, and Option D is incorrect.

Conclusion

After analyzing all the options, only Option B (9, 12, and 15 units) represents a triangle that is similar but not congruent to extit{triangle ABC}. The sides are in the same proportion (3:4:5), but the triangle is scaled up by a factor of 3, making it similar but not identical to the original triangle. Understanding similarity and congruence is fundamental in geometry, allowing us to solve various problems related to shapes and proportions.

In summary, when determining if triangles are similar, always check if their corresponding sides are in proportion. If the sides are in the same ratio but not equal in length, the triangles are similar but not congruent. This concept is essential not only in academic mathematics but also in real-world applications such as architecture, engineering, and design.

Find side lengths of a triangle similar to triangle ABC with vertices A(1,-2), B(1,1), and C(5,-2), but not congruent.