Area Of Right Triangle: Multiples Of 3 Sides
Hey guys! Ever wondered how to calculate the area of a right triangle when its sides are consecutive multiples of 3? It might sound a bit tricky at first, but trust me, it's actually quite straightforward. In this article, we'll break down the process step by step, making it super easy to understand. So, grab your thinking caps, and let's dive in!
Understanding the Basics
Before we jump into the calculations, let's quickly recap some fundamental concepts. First, what exactly is a right triangle? A right triangle is simply a triangle that has one angle measuring exactly 90 degrees. This angle is often called a right angle, hence the name. The side opposite the right angle is the longest side and is known as the hypotenuse. The other two sides are called legs, or sometimes, the cathetus.
Now, how do we find the area of any triangle? The general formula for the area of a triangle is:
Area = (1/2) * base * height
Where the 'base' is any side of the triangle, and the 'height' is the perpendicular distance from the base to the opposite vertex (corner). For a right triangle, it gets even simpler because the two legs themselves can be considered as the base and the height. This is because they are already perpendicular to each other.
So, for a right triangle, the formula simplifies to:
Area = (1/2) * leg1 * leg2
Keep this formula in mind, as we'll be using it extensively throughout our calculations. Next, let's clarify what we mean by "consecutive multiples of 3".
Consecutive Multiples of 3: What Does It Mean?
When we talk about "consecutive multiples of 3", we're referring to numbers that are obtained by multiplying 3 by consecutive integers. Think of the multiplication table of 3 – that’s exactly what we’re talking about! For example, 3, 6, 9, 12, 15, and so on, are all multiples of 3. If we pick three consecutive numbers from this sequence, say 3, 6, and 9, we have ourselves three consecutive multiples of 3.
In the context of our triangle problem, this means the lengths of the sides of the right triangle are three such numbers. So, we can represent the sides as 3x, 3(x+1), and 3(x+2), where x is a positive integer. The challenge now is to figure out which of these sides are the legs and which one is the hypotenuse.
Identifying the Sides of the Right Triangle
Remember the Pythagorean theorem? It’s a crucial concept when dealing with right triangles. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). Mathematically, it's expressed as:
a² + b² = c²
Where:
- a and b are the lengths of the legs
- c is the length of the hypotenuse
Since the hypotenuse is always the longest side, in our case, 3(x+2) must be the hypotenuse. The other two sides, 3x and 3(x+1), are the legs. Now, we can use the Pythagorean theorem to verify this and to find the value of x.
Applying the Pythagorean Theorem
Let’s plug our side lengths into the Pythagorean theorem:
(3x)² + [3(x+1)]² = [3(x+2)]²
Expanding the squares, we get:
9x² + 9(x² + 2x + 1) = 9(x² + 4x + 4)
Now, let's simplify the equation:
9x² + 9x² + 18x + 9 = 9x² + 36x + 36
Combining like terms and rearranging, we have:
9x² - 18x - 27 = 0
We can simplify this further by dividing the entire equation by 9:
x² - 2x - 3 = 0
Solving the Quadratic Equation
We now have a quadratic equation. There are several ways to solve it, such as factoring, completing the square, or using the quadratic formula. Let's use factoring since this equation factors nicely. We need to find two numbers that multiply to -3 and add up to -2. These numbers are -3 and 1. So, we can factor the equation as follows:
(x - 3)(x + 1) = 0
This gives us two possible solutions for x:
x = 3 or x = -1
Since the length of a side cannot be negative, we discard the solution x = -1. Thus, x = 3 is the valid solution. Now that we know the value of x, we can determine the lengths of the sides of our right triangle.
Determining the Side Lengths
We found that x = 3. So, the lengths of the sides are:
- Leg 1: 3x = 3 * 3 = 9
- Leg 2: 3(x+1) = 3 * (3+1) = 3 * 4 = 12
- Hypotenuse: 3(x+2) = 3 * (3+2) = 3 * 5 = 15
So, our right triangle has sides of lengths 9, 12, and 15. Notice that these are indeed consecutive multiples of 3. We can double-check that this is a valid right triangle by plugging these values back into the Pythagorean theorem:
9² + 12² = 81 + 144 = 225
15² = 225
Since 9² + 12² = 15², our triangle satisfies the Pythagorean theorem, confirming it's a right triangle.
Calculating the Area
Now comes the final step: calculating the area of the right triangle. Remember the formula for the area of a right triangle?
Area = (1/2) * leg1 * leg2
We have leg1 = 9 and leg2 = 12. Plugging these values into the formula, we get:
Area = (1/2) * 9 * 12
Area = (1/2) * 108
Area = 54
So, the area of the right triangle is 54 square units. Awesome!
Conclusion
Calculating the area of a right triangle with sides as consecutive multiples of 3 involves a few steps, but it's totally manageable. We started by understanding the basics of right triangles and the Pythagorean theorem. Then, we identified the sides of the triangle using the concept of consecutive multiples of 3. We applied the Pythagorean theorem to find the value of x, which allowed us to determine the side lengths. Finally, we used the formula for the area of a right triangle to calculate the area.
So, there you have it! Next time you encounter a similar problem, you'll know exactly how to tackle it. Keep practicing, and you'll become a pro at solving these types of problems. Keep up the great work, guys! You got this! Remember, math can be fun when you break it down step by step. And always, always double-check your work! You never know when a small mistake might creep in. Happy calculating!