Find the possible values of x.

Attachment 20964

I got 6, but the answer is 12. Why is that? My teachers said there was an easy way to do it without quadratics but I don't remember.

- Feb 26th 2011, 05:44 AMLizarrdoFind the possible values of X using "Similar right triangles"?
Find the possible values of x.

Attachment 20964

I got 6, but the answer is 12. Why is that? My teachers said there was an easy way to do it without quadratics but I don't remember. - Feb 26th 2011, 06:20 AMBAdhi
there are three right angled triangles that you can use to apply phythagoras theorem. apply that theorem to two of them find the lenght of unknown sides with the variable x.

then apply the pythagoras theorem to the other triangle to find x - Feb 26th 2011, 06:24 AMSambit
If you want to avoid quadratics, try to use the fact that the sides of the opposite of the same angles of two similar triangles are proportional in length.

- Feb 26th 2011, 06:30 AMLizarrdo
- Feb 26th 2011, 06:31 AMSoroban
Hello, Lizarrdo!

Quote:

$\displaystyle \text}Find the possible values of }x.$

Code:`C`

*

*| *

* | * x

* |h *

* | *

* | *

A * * * * * * * B

x-5 D x-3

Note that $\displaystyle x > 5.$

We have right triangle $\displaystyle ABC,\:\angle C = 90^o.$

The altitude to the hypotenuse is: .$\displaystyle h = CD.$

It divides the hypotenuse into two segments: .$\displaystyle AD = x-5,\;DB = x-3.$

Fact: The altitude to the hypotenuse is the geometric mean

. . . . . . . . . of the segments of the hypotenuse.

Hence, we have: .$\displaystyle h^2 \:=\:(x-3)(x-5)$ .[1]

In right triangle $\displaystyle CDB\!:\;h^2 + (x-3)^2 \:=\:x^2 \quad\Rightarrow\quad h^2 \:=\:x^2-(x-3)^2$ .[2]

Equate [1] and [2]: .$\displaystyle (x-5)(x-3) \:=\:x^2 - (x-3)^2$

We have: .$\displaystyle x^2 -14x + 24 \:=\:0 \quad\Rightarrow\quad (x-2)(x-12) \:=\:0$

. . Therefore: .$\displaystyle x \,=\,12$