In The Adjacent Figure, Triangle ABC Is Represented, Which Is Right-angled At A, With The Measure Of Angle B Equal To 60°, Point D Being The Midpoint Of Side BC, And AC = 8√3 Cm. The Distance From Point C To Line AD Is Equal To: a) 4 Cm; b) 4√2 Cm; c) 4√3 Cm.

Geometry, a cornerstone of mathematics, involves the study of shapes, sizes, relative positions of figures, and the properties of space. Within this field, triangles hold a special place due to their fundamental nature and the rich set of theorems and properties associated with them. This article delves into a specific geometric problem involving a right-angled triangle, exploring concepts such as angle measures, side lengths, midpoints, and distances. Our main objective is to determine the distance from a specific point to a line within a triangle. We will employ a combination of geometric principles, trigonometric relationships, and algebraic techniques to dissect the problem and arrive at the solution. By understanding the intricacies of this problem, we not only enhance our geometric intuition but also gain valuable skills in problem-solving and analytical thinking.

At the heart of our discussion is the challenge of finding the distance from point C to the line AD in a right-angled triangle ABC. This problem is not just an academic exercise; it represents a common type of question encountered in geometry, with real-world applications in fields like engineering, architecture, and computer graphics. To effectively tackle this problem, we will need to leverage our knowledge of triangle properties, including the Pythagorean theorem, trigonometric ratios, and the properties of special right triangles (such as 30-60-90 triangles). Additionally, understanding the concept of the distance from a point to a line—defined as the length of the perpendicular segment from the point to the line—is crucial. By systematically analyzing the given information and applying relevant geometric principles, we can unravel the complexities of this problem and arrive at a precise and accurate solution. The journey through this geometric exploration will not only provide an answer but also deepen our appreciation for the elegance and power of geometric reasoning.

The problem at hand presents us with a triangle ABC that is right-angled at A. This crucial piece of information immediately allows us to invoke the Pythagorean theorem and trigonometric relationships. We are also told that the measure of angle B is 60 degrees. Given that the sum of angles in a triangle is 180 degrees, and angle A is 90 degrees, we can deduce that angle C is 30 degrees. This makes triangle ABC a 30-60-90 triangle, a special type of right triangle with well-known side length ratios. These ratios are vital for solving the problem efficiently. The next key piece of information is that point D is the midpoint of side BC. This implies that BD = DC, which will be useful when considering lengths and proportions within the triangle. Finally, we are given that AC = 8√3 cm, a concrete measurement that serves as our anchor in this geometric puzzle. The ultimate goal is to find the distance from point C to the line AD, a task that requires us to think strategically about how to relate this distance to the known quantities.

To begin our analysis, let's visualize the triangle and mark the given information. Drawing a clear diagram is often the first step in solving geometry problems, as it allows us to see the relationships between different elements. We can observe that AD is a median of the triangle, connecting vertex A to the midpoint of the opposite side. This median divides the triangle into two smaller triangles, each with its own properties. The distance from C to AD is the length of the perpendicular segment from C to the line containing AD. This perpendicular segment forms another right-angled triangle, which we can use to our advantage. By carefully considering the angles and side lengths within these triangles, we can establish relationships that lead us to the desired distance. The 30-60-90 triangle ratios will play a significant role here, as they allow us to express the side lengths in terms of each other. Additionally, we may need to employ trigonometric functions such as sine, cosine, and tangent to relate angles and side lengths. The combination of these tools will enable us to systematically break down the problem and arrive at the solution.

Let's start by leveraging the properties of the 30-60-90 triangle ABC. In such a triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle. Since AC is opposite the 60-degree angle and has a length of 8√3 cm, we can determine the length of AB, which is opposite the 30-degree angle. Let AB = x. Then, AC = x√3. Setting x√3 = 8√3, we find that x = 8 cm. Therefore, AB = 8 cm. Now, we can find the length of the hypotenuse BC using the Pythagorean theorem or the 30-60-90 triangle ratios. BC = 2 * AB = 2 * 8 = 16 cm.

Since D is the midpoint of BC, BD = DC = BC / 2 = 16 / 2 = 8 cm. Now, consider triangle ADC. We know AC = 8√3 cm and DC = 8 cm. We can use the Pythagorean theorem to find the length of AD: AD² = AC² + DC² = (8√3)² + 8² = 192 + 64 = 256. Taking the square root, we get AD = 16 cm. Next, we need to find the distance from C to AD. Let's call the point where the perpendicular from C meets AD as E. So, we are looking for the length of CE. The area of triangle ADC can be calculated in two ways: using AC as the base and DC as the height, or using AD as the base and CE as the height. The area of triangle ADC is (1/2) * AC * AB = (1/2) * 8√3 * 8 = 32√3 cm². Also, the area of triangle ADC is (1/2) * AD * CE. Setting these two expressions for the area equal to each other, we have (1/2) * 16 * CE = 32√3. Simplifying, we get 8 * CE = 32√3. Dividing both sides by 8, we find CE = 4√3 cm. Therefore, the distance from point C to the line AD is 4√3 cm.

While we have successfully solved the problem using the 30-60-90 triangle properties and the area of a triangle, it's beneficial to explore alternative approaches for verification and to gain a deeper understanding of the problem. One such approach involves using coordinate geometry. We can place the triangle ABC on a coordinate plane, with A at the origin (0,0), B at (8,0), and C at (0, 8√3). Then, the coordinates of D, the midpoint of BC, can be found using the midpoint formula: D = ((8+0)/2, (0+8√3)/2) = (4, 4√3). The equation of the line AD can be found using the two-point form, knowing the coordinates of A and D. The slope of AD is (4√3 - 0) / (4 - 0) = √3. Since the line passes through the origin, its equation is y = √3x. Now, the distance from point C (0, 8√3) to the line y = √3x can be found using the point-to-line distance formula: distance = |Ax + By + C| / √(A² + B²). Rewriting the equation of the line as √3x - y = 0, we have A = √3, B = -1, and C = 0. Plugging in the coordinates of C, we get distance = |√3(0) - 1(8√3)| / √((√3)² + (-1)²) = | -8√3 | / √(3 + 1) = 8√3 / 2 = 4√3 cm. This confirms our previous result, providing a strong verification of our solution.

Another approach involves using trigonometry and similar triangles. We can drop a perpendicular from C to AD, meeting AD at point E. In triangle CED, angle CDE is equal to angle ADB (vertically opposite angles). We can find the sine of angle ADB using the sine rule in triangle ABD. However, this approach is more complex and requires careful consideration of the angles and side lengths. By comparing the results obtained from different methods, we can increase our confidence in the accuracy of our solution. The agreement between the geometric and coordinate geometry approaches strengthens our understanding of the problem and highlights the versatility of mathematical tools in solving geometric challenges. This multi-faceted approach not only provides an answer but also enriches our problem-solving skills and deepens our appreciation for the interconnectedness of mathematical concepts.

In this exploration of a geometric problem involving a right-angled triangle, we successfully determined the distance from point C to line AD. Our journey involved a careful analysis of the given information, the application of geometric principles such as the Pythagorean theorem and 30-60-90 triangle ratios, and the strategic use of algebraic techniques. We found that the distance from C to AD is 4√3 cm. The problem-solving process was not just about finding an answer; it was about understanding the underlying concepts and developing a systematic approach to tackling geometric challenges.

One of the key takeaways from this problem is the importance of visualization. Drawing a clear and accurate diagram is often the first and most crucial step in solving geometry problems. The diagram allows us to see the relationships between different elements, identify relevant triangles, and formulate a strategy. Another important takeaway is the value of leveraging special triangle properties, such as the ratios in a 30-60-90 triangle. These properties provide shortcuts and simplify calculations. We also learned the importance of considering different approaches to problem-solving. By using both geometric and coordinate geometry methods, we not only verified our solution but also gained a deeper understanding of the problem. This highlights the interconnectedness of mathematical concepts and the power of using multiple perspectives. Finally, this problem reinforces the importance of precision and attention to detail in mathematical reasoning. Each step in the solution process requires careful consideration and accurate calculations. By mastering these skills, we can confidently tackle a wide range of geometric problems and appreciate the elegance and power of mathematical thinking. This exploration has not only provided us with a solution but has also enhanced our geometric intuition and problem-solving abilities, making us more effective and confident mathematicians.