1. ## Geometry problems

Hello everyone! I'm getting stuck at this problem and I dont know how to solve it. I'm just curious about it. Any help will be greatly appreciated

2. ## Re: Geometry problems

Could you resubmit, including a diagram, with labels for the various points so that we can refer to lines and angles ?

3. ## Re: Geometry problems

thanks for your attention, here is the diagram

4. ## Re: Geometry problems

Originally Posted by chubakueno
thanks for your attention, here is the diagram
1. You are dealing with 2 right triangles: $\Delta(ABD)$ and $\Delta(ACE)$.

So the point A is located on a half circle (circle of Thales) over BD and point A is located on a half circle (circle of Thales) over CE.

2. You certainely have made an exact drawing(?). See attachment.

a) The point A is the intersection of the green circle and the circle of Thales over BD. The center of the green circle has the coordinates $\left(\frac32 , \frac32 \right)$ with the radius $\frac32 \sqrt{2}$ (why?)

b) Draw the line AC. Construct a right angle in A on AC. The leg of this right angle intersects the prolonged line BD in E.

3. I've done the construction in a coordinate system so you can read the length of x.

4. The line segment BD is divided harmonically by C (inner point) and E (outer point). According to the harmonic partition you'll get the proportion:

$\frac{BC}{CD}=\frac{BE}{DE}$

$\frac{3}{2}=\frac{5+x}{x}$

Solve for x.

5. ## Re: Geometry problems

No, it was just illustrative, but that's the way the problem was given to me. Thank you!

6. ## Re: Geometry problems

My solution is rather more workmanlike !
Start with the right angled triangle ABD and let the side AB = a.
Then
$a^{2}+AD^{2}=5^{2},$ so $AD = \sqrt{25-a^{2}}.$
Now let the angle ACD = $\theta,$ and use the sine rule in each of the triangles ABC and ACD.
From ACD,
$\frac{2}{\sin 45}=\frac{\sqrt{25-a^{2}}}{\sin \theta},$
so
$\sin \theta = \frac{\sin 45\sqrt{25-a^{2}}}{2}.$
From ABC,
$\frac{a}{\sin(180-\theta)}=\frac{3}{\sin 45},$
so
$\sin \theta=\frac{a\sin 45}{3}.$
Equate the two expessions for $\sin \theta,$ simplify, and you find that $a=\frac{15}{\sqrt{13}},$
and from which it follows that AD = $10/\sqrt{13}.$
Call the angle ABD $\phi,$ then from the triangle ABD, $\tan \phi=\frac{10\sqrt{13}}{15\sqrt{13}}=\frac{2}{3},$ and from that you can deduce that $\sin \phi=2/\sqrt{13}$ and $\cos \phi=3/\sqrt{13}.$
Now, finally, turn your attention to the triangle at the other end, ADE.
The angle AED will equal $45 - \phi,$ so making use of the sine rule again,
$\frac{AD}{\sin (45-\phi)}=\frac{x}{\sin 45},$
so
$x = \frac{AD\sin 45}{\sin (45 - \phi)}=\frac{10\sin 45}{\sqrt{13}(\sin 45 \cos \phi-\cos 45 \sin \phi)}.$
$\sin 45 = \cos 45,$ so after cancelling and then substituting for $\cos \phi$ and $\sin \phi,$
we finish up with
$x = \frac{10}{\sqrt{13}(3/\sqrt{13}-2/\sqrt{13})}=10.$