What is the length of median of equilateral triangle?

Decoding the Equilateral Triangle’s Median: It’s Simpler Than You Think!

Equilateral triangles – those perfectly symmetrical shapes with all sides equal – they’ve always held a certain fascination, haven’t they? And nestled within these triangles lies a neat little line called the median. But what is its length, really? Let’s break it down without the jargon.

First off, what’s a median anyway? Simply put, it’s a line drawn from one corner (vertex, if we want to get technical) of a triangle to the exact middle of the opposite side. Every triangle has three of these, one from each corner. And guess what? They all meet at a single point inside – the centroid. Pretty cool, right?

Now, equilateral triangles are special. Because all their sides and angles are equal (60 degrees each, if you’re curious), their medians have superpowers. Seriously! In an equilateral triangle:

• All three medians are exactly the same length. No exceptions.
• Each median is also the triangle’s height (or altitude). That means it forms a perfect right angle with the side it bisects.
• And, get this, each median also perfectly chops the corner angle in half. So, a 60-degree angle becomes two neat 30-degree angles.

Because the median, height, and angle bisector are all rolled into one in an equilateral triangle, figuring out its length becomes surprisingly easy.

Cracking the Code: Calculating the Median’s Length

Here’s the secret formula. If we call the length of one side of our equilateral triangle “a,” then the length of the median (which we’ll call “m”) is:

m = h = (√3 / 2) * a

Yep, that’s it!

Where does this come from? Remember good ol’ Pythagoras? The median splits the equilateral triangle into two identical right-angled triangles. The side of the original triangle becomes the hypotenuse, half the side becomes one leg, and the median is the other leg. A little Pythagorean theorem magic, and boom, you get the formula.

Let’s try an example: Imagine an equilateral triangle with sides of 10 cm each. The median’s length would be:

m = (√3 / 2) * 10 cm ≈ 8.66 cm

Easy peasy, right?

Why Bother Knowing This?

Okay, so why should you care about the median of an equilateral triangle? Well, it pops up in all sorts of geometry problems. Say you know the size of the circle that perfectly encloses the triangle (the circumradius). You can use that to find the triangle’s side length and then, using our formula, find the median. It’s all connected!

The Bottom Line

The median of an equilateral triangle isn’t just some random line. It’s a key property, intimately linked to the triangle’s side length. And because it’s also the height and angle bisector, calculating its length is a breeze. So, next time you encounter an equilateral triangle, remember that handy formula: m = (√3 / 2) * a. You’ll be surprised how useful it can be!