Question

Question #6b302

Answer 1

The length of the altitude `CD` is `4`.

Explanation:

Your situation looks like this:

There is a right angle between AC and BC and a right angle between AD and CD (and BD and CD, of course).

As we have three right triangles, we can apply the Pythagorean Theorem to all three of them:

[1] `" "AC^2 + BC^2 = AB^2`

[2] `" "AD^2 + CD^2 = AC^2`

[3] `" "BD^2 + CD^2 = BC^2`

Furthermore, you know that:

• `AD` is `12` more than the altitude, so `AD = CD + 12`

• `BD` is `3` less than the altitude, so `BD = CD - 3`

Thus, you have

[4] `" "AD = CD + 12`

[5] `" "BD = CD -3`

Last piece of information is that

[6] `" "AB = AD + BD`

Now, let's try to find `CD` with the help of those 6 equations.

First of all, let's plug [2] and [3] into [1]:

`color(blue)(AC^2)" " + color(green)(BC^2) " "= AB^2`

`=>" "(color(blue)(AD^2 + CD^2)) + (color(green)(BD^2 + CD^2)) = AB^2`

Now, let's use [6] and plug `AD + BD` for `color(brown)(AB)`:

`=>" "(AD^2 + CD^2) + (BD^2 + CD^2) = (color(brown)(AD + BD))^2`

Let's simplify this equation:

`=>" "AD^2 + BD^2 + 2CD^2 = (AD + BD)^2`

Use the formula `(a+b)^2 = a^2 + 2ab + b^2` to expand the right side:

`=>" "AD^2 + BD^2 + 2CD^2 = AD^2 + 2*AD * BD + BD^2`

`=>" "cancel(AD^2) + cancel(BD^2) + 2CD^2 = cancel(AD^2) + 2*AD * BD + cancel(BD^2)`

`=> " " 2 * CD^2 = 2 * AD * BD`

Divide both sides by `2`...

`=> " " CD^2 = AD * BD`

Now, we can use [4] and [5]: plug `CD + 12` for `AD` and `CD - 3` for `BD`:

`=> " " CD^2 = color(orange)(AD) * color(purple)(BD)`

`=> " " CD^2 = (color(orange)(CD + 12)) * (color(purple)(CD - 3))`

Expand the right side:

`=> " " CD^2 = CD^2 + 9 CD - 36`

Subtract `CD^2` from both sides...

`=> " " 0 = 9CD - 36`

Solve for `CD`:

`=> " " CD = 4`

Thus, the altitude is `4`, and we can also compute all the other sides from the triangle:

• `AD = 4 + 12 = 16`
• `BD = 4 - 3 = 1`
• `AC = sqrt(AD^2 + CD^2) = sqrt(16^2 + 4^2) = sqrt(272)`
• `BC = sqrt(BD^2 + CD^2) = sqrt(1^2 + 4^2) = sqrt(17)`
• `AB = AD + BD = 16 + 1 = 17`

or

• `AB = sqrt(AC^2 + BC^2) = sqrt(272+17) = 17`