Associate Each Letter From Column A With The Number From Column B So That The Associations Made Express True Statements. In Right Triangle ABC With Angle A = 90 Degrees, AB = 30 Cm, BC = 50 Cm, And AD Perpendicular To BC, D Is On BC.

In the realm of geometry, problems often require us to connect different elements to form a coherent and true statement. This is particularly evident in problems involving triangles, where sides, angles, and altitudes interact in specific ways. Let's delve into a problem that exemplifies this concept, focusing on a right-angled triangle and its properties. This article aims to provide a comprehensive understanding of how to approach and solve such problems, enhancing your problem-solving skills in geometry.

Understanding the Problem

Before we dive into the solution, it's crucial to understand the problem statement thoroughly. We are given a right-angled triangle ABC, with the right angle at vertex A. The sides AB and BC are given as 30 cm and 50 cm, respectively. Additionally, we have an altitude AD drawn from vertex A perpendicular to the hypotenuse BC. The task is to match letters from column A with numbers from column B to form true statements. This type of problem tests our understanding of various geometric properties and theorems related to right-angled triangles, including the Pythagorean theorem, similarity of triangles, and relationships involving altitudes.

The challenge lies in identifying the correct numerical values that correspond to specific lengths or ratios within the triangle. To solve this, we will need to apply our knowledge of geometric principles and perform calculations to determine the missing lengths and ratios. This process will not only help us match the letters and numbers correctly but also deepen our understanding of the relationships within right-angled triangles.

Solving the Geometric Puzzle

To tackle this problem effectively, we need to break it down into smaller, manageable steps. Here’s a structured approach to solve the matching problem:

1. Applying the Pythagorean Theorem

Our first step involves using the Pythagorean theorem to find the length of the remaining side, AC. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In ${ \triangle ABC }$, this translates to:

${ BC^2 = AB^2 + AC^2 }$

We are given AB = 30 cm and BC = 50 cm. Plugging these values into the equation, we get:

${ 50^2 = 30^2 + AC^2 }$

${ 2500 = 900 + AC^2 }$

${ AC^2 = 1600 }$

Taking the square root of both sides, we find:

${ AC = 40 \text{ cm} }$

Therefore, we have determined the length of side AC, which is a crucial piece of information for further calculations.

2. Utilizing Triangle Similarity

Next, we leverage the concept of triangle similarity. In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse divides the triangle into two smaller triangles that are similar to the original triangle and to each other. This means that ${ \triangle ABC }$, ${ \triangle ABD }$, and ${ \triangle CAD }$ are all similar.

From the similarity of triangles, we can establish proportions between corresponding sides. For instance, considering the similarity between ${ \triangle ABD }$ and ${ \triangle ABC }$, we can write:

${ \frac{AD}{AC} = \frac{AB}{BC} }$

We know AB = 30 cm, BC = 50 cm, and AC = 40 cm. Plugging these values in, we get:

${ \frac{AD}{40} = \frac{30}{50} }$

${ AD = \frac{30 \times 40}{50} = 24 \text{ cm} }$

Thus, we have found the length of the altitude AD, which is another key value for our problem.

3. Calculating BD and CD

Now, let's find the lengths of BD and CD. We can use the Pythagorean theorem in ${ \triangle ABD }$ and ${ \triangle CAD }$ to achieve this. In ${ \triangle ABD }$:

${ AB^2 = AD^2 + BD^2 }$

${ 30^2 = 24^2 + BD^2 }$

${ 900 = 576 + BD^2 }$

${ BD^2 = 324 }$

${ BD = 18 \text{ cm} }$

Similarly, in ${ \triangle CAD }$:

${ AC^2 = AD^2 + CD^2 }$

${ 40^2 = 24^2 + CD^2 }$

${ 1600 = 576 + CD^2 }$

${ CD^2 = 1024 }$

${ CD = 32 \text{ cm} }$

We have now calculated the lengths of BD and CD, completing our set of necessary values.

4. Matching Letters and Numbers

With all the lengths calculated, we can now confidently match the letters in column A with the numbers in column B. This step involves carefully reviewing the calculated values and associating them with the corresponding segments or ratios described in the problem. The accurate matching will demonstrate a thorough understanding of the geometric relationships within the triangle.

Key Geometric Principles

This problem highlights several key geometric principles that are fundamental to understanding and solving problems involving right-angled triangles. A deep understanding of these principles is crucial for success in geometry.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry, particularly when dealing with right-angled triangles. It provides a direct relationship between the lengths of the sides, allowing us to calculate the length of an unknown side if the other two are known. This theorem is not just a formula; it's a powerful tool for solving a wide range of geometric problems.

Triangle Similarity

Triangle similarity is another essential concept. Similar triangles have the same shape but may differ in size. This means their corresponding angles are equal, and their corresponding sides are in proportion. The concept of similarity allows us to establish ratios between sides of different triangles, which is invaluable for finding unknown lengths and proving geometric relationships.

Altitude on the Hypotenuse

The altitude drawn from the right angle to the hypotenuse in a right-angled triangle has several interesting properties. As we saw in the problem, it divides the triangle into two smaller triangles that are similar to the original triangle and to each other. This creates a cascade of proportional relationships that can be exploited to solve problems. Additionally, the altitude's length is related to the segments it creates on the hypotenuse, providing further avenues for calculation.

Importance of Structured Problem-Solving

This problem demonstrates the importance of a structured approach to problem-solving in geometry. By breaking down the problem into smaller steps, we can systematically apply relevant theorems and concepts to find the solution. This approach not only makes the problem more manageable but also reduces the chances of making errors.

Step-by-Step Approach

A step-by-step approach allows us to focus on one aspect of the problem at a time. In this case, we first used the Pythagorean theorem to find the missing side length, then applied the concept of triangle similarity to find the altitude, and finally calculated the segments on the hypotenuse. This methodical approach ensures that we don't overlook any crucial information and that our calculations are accurate.

Visual Representation

A visual representation, such as a diagram, is incredibly helpful in geometry problems. Drawing a clear and accurate diagram allows us to visualize the relationships between different elements and identify the relevant theorems and concepts. It also serves as a visual aid for tracking our progress and ensuring that our calculations align with the geometric properties.

Review and Verification

Finally, review and verification are essential steps in problem-solving. Once we have a solution, we should always review our work to ensure that it is logical and consistent with the problem statement. This may involve checking our calculations, verifying the geometric relationships, and ensuring that our answer makes sense in the context of the problem.

Conclusion

Matching letters to numbers in geometric problems, such as the one involving the right-angled triangle, requires a solid understanding of geometric principles and a structured problem-solving approach. By applying theorems like the Pythagorean theorem and the concept of triangle similarity, we can systematically calculate unknown lengths and establish relationships between different elements. The key is to break down the problem into smaller steps, visualize the relationships, and carefully review our work.

This type of problem not only enhances our geometric problem-solving skills but also deepens our appreciation for the elegance and interconnectedness of geometric concepts. By mastering these skills, we can confidently tackle more complex problems and unlock the beauty of geometry.

This article has provided a detailed exploration of how to approach and solve a geometric problem involving a right-angled triangle. By understanding the underlying principles and adopting a structured approach, you can enhance your problem-solving skills and excel in geometry. Remember to practice regularly and apply these concepts to various problems to solidify your understanding.