Mastering Triangle Heights: Understanding the Pythagorean Theorem

What is the height of the triangle with a base of 60 feet and a hypotenuse of 100 feet?

• A. 80 feet
• B. 60 feet
• C. 40 feet
• D. 100 feet

Answer: (A)

To find the height of a triangle when given the base and hypotenuse, one can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is one side, the height is the other side, and the base is 60 feet. Let's denote the height of the triangle as 'h'. Using the Pythagorean theorem, we set up the equation: \[ h^2 + \text{base}^2 = \text{hypotenuse}^2 \] \[ h^2 + 60^2 = 100^2 \] \[ h^2 + 3600 = 10000 \] \[ h^2 = 10000 - 3600 \] \[ h^2 = 6400 \] \[ h = \sqrt{6400} = 80 \text{ feet} \] Therefore, the height of the triangle is 80 feet, confirming that this is the correct answer. This understanding illustrates how to apply the Pythagorean theorem effectively to find unknown dimensions in a triangle when two sides are known.

Understanding triangles is a fundamental skill in geometry, and honestly, tackling problems involving right triangles can feel pretty rewarding. Have you ever wondered how to determine the height of a triangle when you know its base and hypotenuse? It's easier than you might think, thanks to the Pythagorean theorem!

Let’s say we have a triangle with a base of 60 feet and a hypotenuse of 100 feet. What’s the height? At first glance, it might seem tricky, but the solution can be uncovered with a bit of straightforward math. So, how do you access this crucial information? Well, here’s the key: the Pythagorean theorem.

This theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides in a right triangle. In simpler terms, if you know two sides, you can easily find the third. In our case, we have the hypotenuse (100 feet) and the base (60 feet), and we're trying to find the height, which we’ll call 'h.'

Here’s how the math breaks down:

[ h^2 + \text{base}^2 = \text{hypotenuse}^2 ]

[ h^2 + 60^2 = 100^2 ]

Breaking that down further:

[ h^2 + 3600 = 10000 ]

Rearranging the equation gives us:

[ h^2 = 10000 - 3600 ]

So we end up with:

[ h^2 = 6400 ]

And finally:

[ h = \sqrt{6400} = 80 \text{ feet} ]

Voilà! The height of our triangle is 80 feet. This process demonstrates how employing the Pythagorean theorem effectively reveals unknown dimensions in triangles, making geometry feel more approachable.

It’s fascinating how geometry isn’t just about shapes and formulas; it’s about understanding relationships between different elements. Be it in architecture, art, or nature, these triangles pop up everywhere, guiding structures and designs.

So, the next time you encounter a triangle in your studies, you’ll not only remember how to calculate its height, but you’ll also appreciate the deeper connections geometry has with the world around us. Remember, practice makes perfect, and honing these skills will go a long way in building your confidence in geometry.