Solving For AD Length In A Right Triangle ABC Problem

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In right triangle ABC with altitude BD drawn to hypotenuse AC, given AB = 10 and AC = 25, find the length of AD.

In the realm of geometry, right triangles hold a special place, and the relationships within them often lead to elegant solutions. One such relationship arises when an altitude is drawn from the right angle to the hypotenuse. This creates similar triangles, which allows us to use proportions to solve for unknown lengths. In this article, we'll explore how to determine the length of a specific segment within a right triangle using this principle.

Understanding the Problem: Right Triangle ABC with Altitude BD

Let's start by visualizing the scenario. Imagine a right triangle ABC, where angle B is the right angle. Now, draw a line segment BD from vertex B perpendicular to the hypotenuse AC. This line segment, BD, is the altitude to the hypotenuse. We are given that the length of side AB is 10 units and the length of the hypotenuse AC is 25 units. Our mission is to find the length of segment AD. This problem is a classic example of applying the properties of similar triangles formed by the altitude to the hypotenuse of a right triangle. The key here is to recognize that the altitude divides the original triangle into two smaller triangles that are similar to each other and also similar to the original triangle. This similarity allows us to set up proportions between corresponding sides and solve for the unknown length.

Key Concepts and Theorems

Before we dive into the solution, let's briefly review the key concepts and theorems that will underpin our approach:

  • Right Triangle: A triangle with one angle measuring 90 degrees.
  • Hypotenuse: The side opposite the right angle in a right triangle; it's the longest side.
  • Altitude: A line segment from a vertex of a triangle perpendicular to the opposite side (or its extension).
  • Similar Triangles: Triangles with the same shape but potentially different sizes. Their corresponding angles are equal, and their corresponding sides are in proportion.
  • Geometric Mean Theorem: In a right triangle, the altitude to the hypotenuse divides the triangle into two triangles that are similar to the original triangle and to each other. Furthermore, the altitude is the geometric mean between the two segments of the hypotenuse.
  • Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).

With these concepts in mind, we're ready to tackle the problem at hand.

Setting up the Similar Triangles

As mentioned earlier, the altitude BD divides the right triangle ABC into two smaller triangles: triangle ABD and triangle BCD. Crucially, these triangles are similar to each other and to the original triangle ABC. This similarity stems from the Angle-Angle (AA) similarity postulate, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Let's break down why these triangles are similar:

  1. Triangle ABD and Triangle ABC:
    • Both triangles share angle A.
    • Both triangles have a right angle (angle ADB in triangle ABD and angle ABC in triangle ABC).
    • Therefore, by AA similarity, triangle ABD is similar to triangle ABC.
  2. Triangle BCD and Triangle ABC:
    • Both triangles share angle C.
    • Both triangles have a right angle (angle BDC in triangle BCD and angle ABC in triangle ABC).
    • Therefore, by AA similarity, triangle BCD is similar to triangle ABC.
  3. Triangle ABD and Triangle BCD:
    • Since both triangle ABD and triangle BCD are similar to triangle ABC, they are also similar to each other.

This similarity is the cornerstone of our solution. Because the triangles are similar, their corresponding sides are proportional. This means we can set up ratios between the sides of the triangles and use them to solve for the unknown length AD.

Using Proportions to Find AD

Now that we've established the similarity of the triangles, we can set up a proportion involving the sides we know (AB and AC) and the side we want to find (AD). Since triangle ABD is similar to triangle ABC, we can write the following proportion:

AD / AB = AB / AC

This proportion states that the ratio of AD to AB in triangle ABD is equal to the ratio of AB to AC in triangle ABC. This relationship arises directly from the definition of similar triangles – their corresponding sides are in proportion. Notice how AB appears in both the numerator of one ratio and the denominator of the other. This is because AB is a side in both triangles, albeit in different positions relative to the angles.

We are given that AB = 10 and AC = 25. Substituting these values into the proportion, we get:

AD / 10 = 10 / 25

To solve for AD, we can cross-multiply:

25 * AD = 10 * 10

25 * AD = 100

Now, divide both sides by 25:

AD = 100 / 25

AD = 4

Therefore, the length of segment AD is 4 units. This is our solution.

Alternative Approach: Geometric Mean Theorem

While we successfully used proportions derived from similar triangles to find AD, there's another elegant approach using the Geometric Mean Theorem. This theorem provides a direct relationship between the altitude to the hypotenuse and the segments it creates on the hypotenuse.

The Geometric Mean Theorem states that in a right triangle, the altitude to the hypotenuse is the geometric mean between the two segments of the hypotenuse. In our case, this translates to:

BD² = AD * DC

However, this formula directly involves BD and DC, which we don't know yet. But, there's another part of the Geometric Mean Theorem that's even more relevant to our problem:

AB² = AD * AC

This version states that the square of the length of one leg (AB) is equal to the product of the length of the adjacent segment on the hypotenuse (AD) and the length of the entire hypotenuse (AC). This is precisely what we need to solve for AD directly!

Let's plug in the given values: AB = 10 and AC = 25:

10² = AD * 25

100 = AD * 25

Now, divide both sides by 25:

AD = 100 / 25

AD = 4

As you can see, we arrive at the same answer (AD = 4) using the Geometric Mean Theorem. This demonstrates the power and interconnectedness of geometric principles. This alternative approach offers a more direct route to the solution, highlighting the elegance of geometric theorems.

Conclusion: AD = 4 Units

In conclusion, by leveraging the properties of similar triangles formed by the altitude to the hypotenuse in a right triangle, or by applying the Geometric Mean Theorem, we have successfully determined that the length of segment AD is 4 units. This problem showcases the beauty and utility of geometric relationships, providing a solid foundation for tackling more complex geometric challenges. Whether you choose to use proportions derived from similar triangles or the Geometric Mean Theorem, the key is to understand the underlying principles and apply them strategically. The ability to recognize similar triangles and their properties is a valuable skill in geometry, and mastering these concepts will undoubtedly enhance your problem-solving abilities in various mathematical contexts. The solution not only provides a numerical answer but also reinforces the importance of understanding and applying fundamental geometric theorems. Remember, geometry is not just about memorizing formulas; it's about developing a spatial intuition and the ability to see relationships between shapes and figures. This problem serves as a great example of how these skills can be applied to solve practical problems in geometry and beyond. The process of solving this problem, from identifying similar triangles to setting up proportions and applying the Geometric Mean Theorem, provides a valuable learning experience that can be applied to a wide range of geometric problems.

By understanding the relationships within right triangles, we unlock a powerful toolkit for solving geometric problems. The altitude to the hypotenuse is a common theme in geometry, and mastering the concepts associated with it will prove invaluable in future mathematical endeavors. Remember to always visualize the problem, identify the key relationships, and choose the most efficient method to arrive at the solution. Geometry is a fascinating field, and with practice and understanding, you can conquer even the most challenging problems.