Stewart's theorem

Do you remember when you learned about finding the middle point between two points on a line? You just took the average of the x-values and the average of the y-values, right? Well, what if you had a triangle and you wanted to find the length of a side that wasn't next to the right angle? That's where Stewart's Theorem comes in!

Stewart's Theorem is a formula that helps you find the length of a side in a triangle when you know the lengths of the other two sides and the length of the line segment that divides the opposite side into two smaller segments. Let's break it down into simpler steps:

Step 1: Draw your triangle and label the sides and segments.

Step 2: Pick a side (let's call it side "a") that isn't next to the angle you're interested in (let's call that angle "C").

Step 3: Draw a line segment from angle C to side "a" that divides side "a" into two smaller segments (let's call those segments "m" and "n").

Step 4: Write down the formula for Stewart's Theorem: (m * m * n) + (n * n * m) = (a * a * d) + (b * b * d)

Step 5: Substitute in the values you know: the lengths of sides "b" and "c", the length of segment "d", and the lengths of segments "m" and "n".

Step 6: Use algebra to solve for the missing length (in this case, the length of side "a").

And voila! You now know the length of side "a" in your triangle. Just remember to use this theorem carefully, and always double check your work!