Similar Triangles Finding Side Lengths Of A Non-Congruent Triangle

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In the realm of geometry, similar triangles hold a special place. They are triangles that share the same shape but differ in size. This similarity implies that their corresponding angles are equal, and their corresponding sides are in proportion. However, congruence is a stricter condition. Congruent triangles are not just similar; they are identical in every aspect – size, shape, and angles. Therefore, congruent triangles have the same side lengths, whereas similar triangles only need proportional side lengths.

When we delve into the concept of similar triangles, we quickly encounter the side lengths as a crucial element. The ratios between corresponding sides of similar triangles are constant. This constant ratio is often referred to as the scale factor. To determine if two triangles are similar, comparing the ratios of their sides is a reliable method. If the ratios are equal, the triangles are similar. If, in addition to being similar, the triangles are of different sizes, they are not congruent.

Our problem presents us with triangle ABC, defined by its vertices A(1, -2), B(1, 1), and C(5, -2). To determine the potential side lengths of a similar but non-congruent triangle, we first need to find the side lengths of triangle ABC. We can achieve this by using the distance formula, which calculates the distance between two points in a coordinate plane.

The distance formula is derived from the Pythagorean theorem and is expressed as follows:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.

Let's apply the distance formula to find the side lengths of triangle ABC:

  1. Side AB: Distance between A(1, -2) and B(1, 1)

    AB=(1−1)2+(1−(−2))2=02+32=9=3AB = \sqrt{(1 - 1)^2 + (1 - (-2))^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3

  2. Side BC: Distance between B(1, 1) and C(5, -2)

    BC=(5−1)2+(−2−1)2=42+(−3)2=16+9=25=5BC = \sqrt{(5 - 1)^2 + (-2 - 1)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

  3. Side AC: Distance between A(1, -2) and C(5, -2)

    AC=(5−1)2+(−2−(−2))2=42+02=16=4AC = \sqrt{(5 - 1)^2 + (-2 - (-2))^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4

So, the side lengths of triangle ABC are AB = 3, BC = 5, and AC = 4. Notice that these side lengths satisfy the Pythagorean theorem (3² + 4² = 5²), indicating that triangle ABC is a right-angled triangle. This characteristic will be useful when we evaluate the options.

Now that we know the side lengths of triangle ABC, we can examine the given options to find a set of side lengths that form a similar but non-congruent triangle. Remember, for triangles to be similar, their corresponding sides must be in proportion.

Let's analyze each option:

  • Option A: 3, 4, and 5 units

    These side lengths are the same as triangle ABC. Therefore, this triangle would be congruent to triangle ABC, not just similar. So, option A is incorrect.

  • Option B: 9, 12, and 15 units

    Let's check the ratios of these side lengths to the side lengths of triangle ABC:

    93=3\frac{9}{3} = 3

    124=3\frac{12}{4} = 3

    155=3\frac{15}{5} = 3

    The ratios are equal, with a scale factor of 3. This means a triangle with side lengths 9, 12, and 15 is similar to triangle ABC. Furthermore, it is not congruent because the side lengths are different. Option B is a potential answer.

  • Option C: 6, 4, and 5 units

    Let's check the ratios:

    63=2\frac{6}{3} = 2

    44=1\frac{4}{4} = 1

    The ratios are not equal, so this triangle is not similar to triangle ABC. Option C is incorrect.

  • Option D: 9, 12, and 5 units

    Let's check the ratios:

    93=3\frac{9}{3} = 3

    124=3\frac{12}{4} = 3

    55=1\frac{5}{5} = 1

    The ratios are not equal, so this triangle is not similar to triangle ABC. Option D is incorrect.

After evaluating all the options, we found that option B (9, 12, and 15 units) is the only set of side lengths that form a triangle similar but not congruent to triangle ABC. This is because the side lengths are in proportion to the side lengths of triangle ABC (with a scale factor of 3). Therefore, the correct answer is option B.

In summary, we tackled the problem by first understanding the properties of similar and congruent triangles. We then calculated the side lengths of triangle ABC using the distance formula. Finally, we compared the ratios of the side lengths of the given options to the side lengths of triangle ABC to determine which option represented a similar but non-congruent triangle. This approach allowed us to methodically solve the problem and arrive at the correct answer.

The Vertices of Triangle ABC A(1,-2), B(1,1), and C(5,-2): Finding Similar Yet Non-Congruent Triangle Side Lengths

Let's explore the fascinating world of similar triangles and how their side lengths play a crucial role in determining their relationships. This article will guide you through a step-by-step solution to a geometry problem, focusing on understanding the concepts of similarity and congruence in triangles. We'll break down each step, ensuring you grasp the underlying principles and can confidently apply them to similar problems.

Unpacking the Concept of Similar Triangles

In geometry, similarity is a fundamental concept that describes shapes that are the same but can be different sizes. Imagine shrinking or enlarging a photograph – the image remains the same, but its size changes. This is the essence of similarity. In the context of triangles, similar triangles have the same angles but can have different side lengths. The key characteristic of similar triangles is that their corresponding sides are proportional, meaning the ratio between any two corresponding sides is constant. This constant ratio is often referred to as the scale factor.

On the other hand, congruent triangles are a special case of similar triangles. Congruent triangles are identical in every way – they have the same shape, size, and angles. This means their corresponding sides are not only proportional but also equal in length. In essence, congruent triangles are exact copies of each other.

Understanding the difference between similarity and congruence is crucial for solving geometry problems. While similar triangles share the same shape, congruent triangles are the same shape and size. This distinction affects how we analyze their side lengths and angles.

Analyzing the Given Triangle ABC

Our problem presents us with a specific triangle, triangle ABC, defined by its vertices A(1, -2), B(1, 1), and C(5, -2). To identify a triangle that is similar but not congruent to ABC, our first step is to determine the side lengths of triangle ABC. This will serve as our baseline for comparison.

To calculate the side lengths, we employ the distance formula, a powerful tool derived from the Pythagorean theorem. The distance formula allows us to calculate the straight-line distance between any two points on a coordinate plane. It's expressed as follows:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Here, (x₁, y₁) and (x₂, y₂) represent the coordinates of the two points we want to measure the distance between. Let's put this formula to work to find the lengths of sides AB, BC, and AC of triangle ABC.

Finding the Length of Side AB

Side AB connects points A(1, -2) and B(1, 1). Plugging these coordinates into the distance formula, we get:

AB=(1−1)2+(1−(−2))2=02+32=9=3AB = \sqrt{(1 - 1)^2 + (1 - (-2))^2} = \sqrt{0^2 + 3^2} = \sqrt{9} = 3

Thus, the length of side AB is 3 units.

Finding the Length of Side BC

Side BC connects points B(1, 1) and C(5, -2). Applying the distance formula:

BC=(5−1)2+(−2−1)2=42+(−3)2=16+9=25=5BC = \sqrt{(5 - 1)^2 + (-2 - 1)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

So, the length of side BC is 5 units.

Finding the Length of Side AC

Side AC connects points A(1, -2) and C(5, -2). Using the distance formula:

AC=(5−1)2+(−2−(−2))2=42+02=16=4AC = \sqrt{(5 - 1)^2 + (-2 - (-2))^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4

Therefore, the length of side AC is 4 units.

Now we know the side lengths of triangle ABC are AB = 3, BC = 5, and AC = 4. An interesting observation here is that these lengths satisfy the Pythagorean theorem (3² + 4² = 5²), which tells us that triangle ABC is a right-angled triangle. This characteristic could be useful when we evaluate the answer choices.

Evaluating the Answer Choices for Similarity and Non-Congruence

With the side lengths of triangle ABC determined, we can now analyze the given options to identify a triangle that is similar but not congruent to ABC. Remember, similarity implies proportional sides, while non-congruence means the triangles should not be the same size.

To check for similarity, we'll calculate the ratios of the side lengths of each option to the corresponding side lengths of triangle ABC. If the ratios are equal, the triangles are similar. If the side lengths are different, the triangles are not congruent.

Let's examine each option:

Option A: 3, 4, and 5 units

These side lengths are identical to those of triangle ABC. This means a triangle with these side lengths would be congruent to triangle ABC, not just similar. Since we're looking for a non-congruent triangle, option A is incorrect.

Option B: 9, 12, and 15 units

Let's calculate the ratios of these side lengths to the corresponding sides of triangle ABC:

  • Ratio of the shortest sides: 9 / 3 = 3
  • Ratio of the middle sides: 12 / 4 = 3
  • Ratio of the longest sides: 15 / 5 = 3

All the ratios are equal to 3. This indicates that a triangle with side lengths 9, 12, and 15 is similar to triangle ABC. The scale factor is 3, meaning this triangle is three times larger than triangle ABC. Since the triangles have different sizes, they are not congruent. Option B appears to be a correct answer.

Option C: 6, 4, and 5 units

Let's calculate the ratios:

  • Ratio of the shortest sides: 6 / 3 = 2
  • Ratio of the middle sides: 4 / 4 = 1

The ratios are not equal. Therefore, a triangle with side lengths 6, 4, and 5 is not similar to triangle ABC. Option C is incorrect.

Option D: 9, 12, and 5 units

Let's calculate the ratios:

  • Ratio of the shortest sides: 9 / 3 = 3
  • Ratio of the middle sides: 12 / 4 = 3
  • Ratio of the longest sides: 5 / 5 = 1

The ratios are not equal. Therefore, a triangle with side lengths 9, 12, and 5 is not similar to triangle ABC. Option D is incorrect.

Conclusion: Identifying the Similar Non-Congruent Triangle

After carefully analyzing each option, we've determined that option B (9, 12, and 15 units) is the only set of side lengths that form a triangle similar but not congruent to triangle ABC. The sides are proportional, with a scale factor of 3, but the different sizes ensure non-congruence. This solution highlights the importance of understanding the definitions of similarity and congruence and applying the distance formula to calculate side lengths accurately.

Therefore, the final answer is Option B. This exercise demonstrates how a methodical approach, combined with a solid understanding of geometric principles, can lead to a clear and accurate solution. Remember, the key to solving geometry problems lies in breaking them down into smaller, manageable steps and applying the appropriate formulas and concepts. Happy problem-solving!