Ancient Chinese Problem

First and foremost , Have a look at the bamboo stick . We know that the length of the original unbroken bamboo stick is 10m . So the break and the hypotenuse sum to 10m.

The hypotenuse will obviously be longer than the break so the bamboo stick must have been broken assymetrically (not symmetrically)

Let's call the length of the break as xm. then the hypotenuse automatically becomes (10-x).

So now let's reflect this in the triangle

We can now use the Pythagoras Theorem and solve for x

Hope this helped

Let the height where the stem is broken be $h\ \rm ft$ from the ground. Then the length of the top part of the broken stem is $10-h\ \rm ft$ . By Pythagorean theorem , we have:

$\begin{aligned} h^2 + 3^2 & = (10-h)^2 \\ h^2 + 9 & = h^2 - 20 h +100 \\ 20h & = 91 \\ \implies h & = \frac {91}{20} = \boxed {4.55}\ \rm ft \end{aligned}$

Surely they didn’t use the “ft” in ancient China?

This problem can be solved using the Pythagorean Theorem .

For the triangle made by the bamboo sticks and the ground, let's name the sides $a$ , $b$ , and $c$ . An image is shown for reference at the end of this explanation.

Here's what we know:

• $b=3 \text{ ft}$

• $a + c = 10 \text{ ft}$

The Pythagorean Theorem states that, for a right triangle with side lengths $a$ , $b$ , and $c$ as shown in the attached image:

$a^2 +b^2 = c^2$

In order to solve the above equation, we need to rewrite $a$ in terms of $c$ , or vice-versa. I chose to solve for $a$ in terms of $c$ ..

From $a+c=10$ , we can deduce that $a=10-c$ . So, plugging this as well as $b=3$ into the formula gives:

$(10-c)^2+(3)^2=c^2$

This comes out to $c=109/20=5.45$ .

So, now that we know the values of $c$ and $b$ , we can solve for $a$ .

If $a=10-c$ , then $a=10-5.45$ , which means $a=4.55$ .

So, the height is 4.55 ft .