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How to Find the Legs of a Right Triangle

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Finding the Legs of a Right Triangle

In the world of geometry, right triangles hold a special place. These triangles, defined by their single right angle, are fundamental to understanding many geometric concepts. One common problem involves finding the lengths of the legs of a right triangle when you know the lengths of the hypotenuse and one leg. This is where the power of similar triangles and the geometric mean come into play.

Understanding Similar Triangles

Similar triangles are triangles that have the same shape but different sizes. Their corresponding angles are equal, and their corresponding sides are proportional. This concept is crucial for solving problems involving right triangles.

The Geometric Mean

The geometric mean is a special type of average used to relate the sides of similar triangles. In a right triangle, when you draw an altitude from the right angle to the hypotenuse, it divides the triangle into three similar triangles. The geometric mean theorem states that the altitude is the geometric mean of the two segments of the hypotenuse. In other words, the altitude squared equals the product of the two segments.

Finding the Legs

Let's consider a right triangle ABC, where the right angle is at C. The hypotenuse is AB, and we know the lengths of AC (one leg) and AB (the hypotenuse). To find the length of BC (the other leg), we can follow these steps:

  1. Draw the Altitude: Draw an altitude from the right angle C to the hypotenuse AB. This altitude intersects AB at point D.
  2. Identify Similar Triangles: Notice that triangle ABC is similar to triangle ACD and triangle CBD. These three triangles are all similar because they share the same angles.
  3. Apply the Geometric Mean: We know that CD is the geometric mean of AD and DB. Therefore, CD2 = AD * DB.
  4. Solve for the Missing Segment: We can use the Pythagorean theorem to find the lengths of AD and DB. For example, in triangle ACD, AC2 = AD2 + CD2. We can then solve for AD. Similarly, we can find DB in triangle CBD.
  5. Find the Other Leg: Now that we know the lengths of AD and DB, we can find the length of BC using the Pythagorean theorem in triangle ABC. BC2 = AB2 - AC2.

Example

Let's say we have a right triangle ABC with AC = 5, AB = 13, and we want to find BC. Following the steps above:

  1. Draw the altitude CD.
  2. Identify similar triangles: ABC, ACD, and CBD.
  3. Apply the geometric mean: CD2 = AD * DB.
  4. Solve for AD and DB: Using the Pythagorean theorem, we find AD = 12 and DB = 1.
  5. Find BC: BC2 = 132 - 52 = 144. Therefore, BC = 12.

Conclusion

Finding the legs of a right triangle using similar triangles and the geometric mean is a powerful technique in geometry. By understanding these concepts, you can solve a variety of problems involving right triangles and deepen your understanding of their properties. Remember to carefully identify the similar triangles, apply the geometric mean theorem, and use the Pythagorean theorem to find the missing side lengths.

Pythagorean Theorem