If A 49 And A 10 Find C

Finding the Hypotenuse: Solving for c When the Legs are 49 and 10

Imagine you're a carpenter laying out a perfect right angle for a new deck, or a surveyor calculating the direct distance across a plot of land. In both cases, you’re dealing with a right triangle—a triangle with one exact 90-degree angle. The two shorter sides that form this right angle are called the legs, and the longest side, directly opposite the right angle, is the hypotenuse. A classic and powerful tool for this job is the Pythagorean theorem, a cornerstone of geometry for over 2,500 years. This article will walk you through the complete process of finding the length of the hypotenuse, denoted as c, when the two legs measure 49 units and 10 units. We will move from the simple formula to the exact calculation, explore the profound "why" behind the theorem, and address common questions that arise when applying this fundamental principle.

The Step-by-Step Calculation: From Formula to Answer

The Pythagorean theorem states a simple, elegant relationship for any right triangle: the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). The formula is: a² + b² = c²

In our specific problem, we are given the two legs. Let’s assign:

• a = 49
• b = 10

Our goal is to solve for c.

Step 1: Square each leg. First, calculate the square of 49 and the square of 10.

• 49² = 49 × 49 = 2,401
• 10² = 10 × 10 = 100

Step 2: Sum the squares. Add the results from Step 1 together.

• 2,401 + 100 = 2,601 This sum, 2,601, represents c².

Step 3: Find the square root to solve for c. To isolate c, we take the square root of both sides of the equation.

• c = √2,601

Step 4: Simplify the square root. We need to find a number that, when multiplied by itself, equals 2,601. Through calculation or recognition, we find:

• 51 × 51 = 2,601 Therefore:
• c = 51

Conclusion of Calculation: When the legs of a right triangle are 49 units and 10 units, the length of the hypotenuse c is exactly 51 units. This is a classic example of a Pythagorean triple—a set of three positive integers (49, 10, 51) that satisfy the theorem. While the most famous triple is (3, 4, 5), this set demonstrates that the theorem works perfectly with much larger numbers.

The Science Behind the Theorem: Why Does This Work?

The Pythagorean theorem is not just a memorized formula; it is a geometric truth with multiple proofs. One of the most intuitive involves area.

Consider constructing a large square. On each of its four sides, attach an identical right triangle with legs of length a (49) and b (10). The hypotenuses of these four triangles form a smaller, tilted square in the center. The area of the large outer square can be expressed in two ways:

1. As the side length squared: (a + b)².
2. As the sum of the areas of the four triangles and the inner square: 4 × (½ab) + c².

Setting these equal: (a + b)² = 4 × (½ab) + c² Expanding the left side: a² + 2ab + b² = 2ab + c² Subtracting 2ab from both sides yields the famous: a² + b² = c²

This proof shows that the relationship is fundamentally about conservation of area. The area "lost" from the corners of the large square (the four triangular sections) is exactly equal to the area gained in the central square (c²). For our numbers (49 and 10), the theorem guarantees that the combined area of squares built on the 49-unit and 10-unit legs will always equal the area of a square built on the 51-unit hypotenuse, a relationship that holds true for every possible right triangle.

Frequently Asked Questions (FAQ)

Q1: Does the order of 49 and 10 matter? A: No. The theorem is symmetric: a² + b² = b² + a². Whether you assign a=49 and b=10, or a=10 and b=49, the sum of their squares is identical (2,401 + 100 = 100 + 2,401 = 2,601). The hypotenuse c will be 51 in either case.

Q2: What if the triangle isn't a right triangle? A: The Pythagorean theorem is only valid for right triangles. If a triangle has sides 49, 10, and 51 but is not a right triangle, the theorem will not hold. You can test for a right angle: if a² + b² = c² (with c being the longest side), the triangle must be a right triangle. Here, 49² + 10² = 2,401 + 100 = 2,601, and 51² = 2,601, confirming a right angle between the 49 and 10 sides.

Q3: Could there be a negative solution for c? A: In the context of geometry and physical lengths, no. Length is a scalar quantity with only positive magnitude. While the equation c² = 2,601 has two mathematical solutions (c = 51 and c = -51), the negative value has no physical meaning for the length of a side. We always take the positive square root.

Q4: How do I know which side is the hypotenuse? A: The hypotenuse is always the longest side in a right triangle and is always opposite the 90-degree angle. In our problem, we were told the two shorter sides (49 and 10). The calculated c (