Removing triangels from rectangle

[HellishER]

Hi, i'm placing some stones in my garden, but i need to cut of two triangles from one stone for elevating ground height.
I have drawn a picture to help with the explanation.

The stones/blocks are 25 x 50 cm.
The height difference between block-x and block-y is 8 cm.
I only want to cutaway on block-z, the yellow/black part. But whats the distance of C?
If i could calculate C then i could cut from the corner of the block in a straight line to the C-distance from the opposite corner.

Cheers

 

[Ishuda]

I'm assuming the red outlined blocks top right corner 'coincides' with the left top corner of the right block and the red outlined blocks bottom left corner 'coincides' with the right bottom corner of the right block.

As a practical solution, call the top right hand corner of the red outlined block x. How far is it from the bottom right hand corner of the left hand block to x:
x² = 25² + 50²
or x ~ 55.9
So, inscribe a circle from the lower right hand corner of the left hand block and where it crosses a height of 33 on the right is where the upper left corner of your left hand block will be. Lay out your middle block and do the necessary measurements from there.

Or, since you now know the location of all three blocks, go ahead and compute the distances.

 

[HellishER]

I'm assuming the red outlined blocks top right corner 'coincides' with the left top corner of the right block and the red outlined blocks bottom left corner 'coincides' with the right bottom corner of the right block.

Thats correct.

Sorry, but i don't see how your answer would give me the value of C.

 

[Ishuda]

I'll outline how I would go about it:

Change notation so that a co-ordinate system's origin is at the lower left corner of the left hand block. You know the corners of the right hand block: (0,0), (0,25), (50,0), and (50,25). Thus the lower left corner of the middle block is at (50,0). Call that point (x₀, y₀). Call the upper right corner of the middle block (x₁,y₁). We are given y₁ as 33. The value of x₁ can be determined since the opposite corner lies on a circle of radius 55.9 [see the post above] with center at (50,0) and a y value of y₁. Thus
(x₁-50)² + 33² = 55.9²

Now find the angle a₀ between one side of the middle block (say the upper side) and the line between (x₀,y₀) and (x₁,y₁), i.e.
a₀ = arccos(50/55.9)
Determine the formula for the line between (x₀,y₀) and (x₁,y₁)
y = y₀ + (y₁ - y₀) (x - x₀) / (x₁ - x₀)
or
y = y₀ + c₀ (x - x₀)

There is now another formula which will give the equation for the line of the upper side of the middle block which involves a₀, b₀, and c₀
y = y₁ + c₁ (x - x₁)
I don't remember the formula for c₁ in terms of a₀ and c₀ off the top of my head at the moment so you would have to look it up. Call the upper left corner of the middle block (x₃,y₃) and it satisfies the equation. The distance between (x₁,y₁) and (x₃,y₃) is 50, i.e. (x₃,y₃) also satisfies the equation for the circle
(x - x₁)² + (y - y₁)² = 50²
Using those two equation, the point (x₃,y₃) can be determined.

As I understand the diagram, C is now given by
C = 50 - x₃

See why I said that, as a practical matter, it was easier to just measure the distance?

 

[HellishER]

Ishuda, thats to advance math for me.
Thanks for your time and help. I got out of time and was forced to come up with a method to measure the distance by placing blocks on the ground and do many alignments. Which i really didn't want to do.
Seams when the height difference is 8 or 10 then the forumla is: C = height_distance / 2
As Denis wrote the answer of my question was 4.

Cheers!

 

[Deleted member 4993]

Well, if you got a slightly wider block (25.325 by 50) to be the block in the center,
then your troubles would vanish and everything would "fit cleanly"

The cut to bottom block would be at 80.8 degrees, or starting at 4.052 from top
right corner, cutting towards bottom right corner; similarly with top block.

...and you will live happily forever after...

How do you propose to measure that on cement block - using vernier calipers, and marking those with chalk and cutting those with hammer/chisels??

 

[Ishuda]

Sorry you had to go to the many alignments.

If you need to do this in the future, figure out, according to my first post, the distance you will need to determine the point x (in this case 55.9 inches). Now take a piece of string and tie a nail at one end and drive it into the ground at where the lower right corner of the left hand block would be and then measure out 55.9 inches from there as the length of the string. Swing the string in an arc to cross the line 33 inches (in this case) up from the bottom line defined by the bottom edge of the left hand block. That would be the upper left corner of the left hand block and its top edge would line along that line 33 inches up.

Now you have both the right hand and left hand block in their proper place. You can lay the middle block in its place and do the measurement.