The area of a square with right triangle inside it

In summary, the problem involves finding the area of a square by using basic trigonometric relations. The approach taken was to find the length of BC, which required finding the unknowns of the right triangle CDE. However, the solution was not immediately clear and required using the similarity of the upper two right triangles AED and BDC. After setting up equations for the lengths of BC, DB, and AD, the solution was found.
  • #1
JasonHathaway
115
0

Homework Statement



I need to find the area of the square in the following figure:

image.jpg

Homework Equations


Basic Trig relations.

The Attempt at a Solution


I aimed to find the length of BC, but first I had to find the unknowns of the right triangle CDE, which are EC=5m, <DCE=36.86ْ , <DEC=53.13ْ .

Then I thought that I can use some trig relation for angles to get <BDC and BCD respectively (since the angles of the square are all 90ْ).

And that's where I am stuck. Any help?
 
Last edited:
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  • #2
figure shows as a broken link
 
  • #3
Just fixed it. Sorry about that.
 
  • #4
Hm ... interesting problem. It seems as though there's likely to be some trick to it that you are not seeing, but whatever it is, I'm not seeing it either.
 
  • #5
The upper two right triangles AED and BDC are similar. Look at the angles.

squaretriangle.JPG
 
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  • #6
Call BC = s, DB = y, and AD = s-y. It's easy to get two equations in s and y.
 
  • #7
I got it now. Thank you everyone for your help. I really appreciate it.
 

1. How do you calculate the area of a square with a right triangle inside it?

To calculate the area of a square with a right triangle inside it, you can use the formula A = (s^2)/2, where A is the area of the square and s is the length of the hypotenuse of the right triangle.

2. What is the relationship between the square and the right triangle in this scenario?

The square and the right triangle in this scenario are complementary shapes, meaning they fit perfectly together to form a larger shape. The right triangle is inscribed inside the square, with its hypotenuse being the diagonal of the square.

3. Can the area of the square and the right triangle be equal?

No, the area of the square and the right triangle cannot be equal. The area of a square is always greater than the area of a right triangle, as the square encompasses the triangle and has extra space around it.

4. How does the size of the right triangle affect the area of the square?

The size of the right triangle affects the area of the square in that the larger the right triangle, the larger the area of the square will be. This is because the length of the hypotenuse (which determines the size of the triangle) is used in the formula to calculate the area of the square.

5. Can this concept be applied to other shapes besides a square and a right triangle?

Yes, this concept can be applied to other shapes as well. Any two complementary shapes can be used to calculate the area of a larger shape. For example, a circle and a square can be used to calculate the area of a circle inscribed inside a square.

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