Question

Given the right triangle ABC with altitude BD drawn to the hypotenuse AC. If AC=6 and DC=4, what is the length of BC in simplest radical form ?

Answer 1

This problem is an application of the Geometric mean theorem. It says that

`(6)/(x)=(x)/(4)`

Comment: In other words, it says that the length of BC (x) is the geometric mean between the lengths of AC and DC.

Then,

`x^2=6\cdot4=24`
`x=\sqrt[]{24}=2\cdot\sqrt[]{6}`

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Let's talk a little about the simplest radical form of a square root

`\sqrt[]{a}`

The first step to finding it is to write the number within the root as a product of prime powers, such product is called its integer factorization. Let's do that for 24:

Then, the integer factorization of 24 is

`24=2^3\cdot3`

Thus,

`\sqrt[]{24}=\sqrt[]{2^3\cdot3}`

The idea now is to take out of the root all we can. The rule is that we can only take out powers of 2 (for our root is a square root). In the expression

`2^3\cdot3`

There is only one power of 2, within 2^3. We can write it as

`2^2\cdot2\cdot3`

How are we going to take out it? We are going to take out the base of the power, which is 2 in this case. Then,

`\sqrt[]{24}=2\cdot\sqrt[]{2\cdot3}=2\cdot\sqrt[]{6}`

In simple terms, the simplest radical form of a root is what results after taking out the root all that can be taken out.