Right triangle with four jointed links

[chawki]

Homework Statement

A right triangle ABCD is formed from four jointed links. The joint D at the midpoint of the hypotenuse is pushed until it lies on the leg AB.

Homework Equations

Find the height x when a = 50 mm

The Attempt at a Solution

give me a hint please cause I'm lost

 

[SteamKing]

Well, if I was given a right triangle and the length of two of the sides, I would first find out what the length of the third side is, Pythagoras.

 

[chawki]

ok, let the distance AB=x

(2a)²=a²+x²
x²= 4a²-a²=3a²
x=\sqrt{}3 * a
x=86.60mm
and then DB=86.60-50
DB=36.6mm.

But then?

 

[Apphysicist]

I would find the angle DCB using the initial right triangle, then note that the new, smaller triangle is isosceles because DC=(2a)/2 and CB=a. Then use the fact that the altitude from C bisects angle DCB and creates two new right-triangles...you then have a relation between angle, height, and hypotenuse.

 

[chawki]

sounds complicated!

 

[chawki]

How can we find the angle DCB ?

 

[chawki]

I would find the angle DCB using the initial right triangle, then note that the new, smaller triangle is isosceles because DC=(2a)/2 and CB=a. Then use the fact that the altitude from C bisects angle DCB and creates two new right-triangles...you then have a relation between angle, height, and hypotenuse.

hello

 

[chawki]

AB-a=86.6-50=36.6mm

now, a²=x²+(36.6/2)²
x=46.53mm ?