Can someone please help me with these three geometry questions??

Question

Dean drew a large right triangle ABC. He found and labeled the midpoint of the hypotenuse X. He drew a line segment YX parallel to line segment BC. His next step was to draw line segment XZ parallel to the line segment AB. He finally drew line segment BX.

 

* What is the length of YX in relationship to BC

 

 

 

*What is the area of triangle AYX in relationship to the area of triangle ABC? Explain how you know

 

 

 

* Prove that the tree triangles below are similar to each other

 

TRIANGLE ABC ~ TRIANGLE AYX ~ TRIANGLE XZC 

Answer 1

First, prepare a right triangle whose hypotenuse is AC with AB being the vertical side and BC the horizontal side. Then do those drawings for XY and XZ. 

Question 1: * What is the length of YX in relationship to BC
Answer:  Thales theorem says that  YX/BC = AX/AC. Since AX is half of the AC, therefore YX = BC/2. 

Question 2: *What is the area of triangle AYX in relationship to the area of triangle ABC? Explain how you know.
Answer: Again, using Thales theorem we will figure out that AY = AB/2 as well. This means that the triangle AYX has vertical and horizontal dimensions that are twice smaller than that of ABC. Since the area of a triangle is found by using a product of the bases: Big Area = 1/2 AB*BC, we get a total of a factor of 4 times smaller area for the smaller triangle. Small Area = 1/2 AY*YX = 1/2 (AB/2)*(BC/2) = 1/2 AB*BC/4 = New Area/4. 

Question 3: * Prove that the tree triangles below are similar to each other.
Answer: Proof can be done using similarity postulates. Good luck! (This needs some drawing, so I leave it for you).