Finding the diagonal of an irregular quadrilateral

  • Thread starter gaobo9109
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That gives two equations in the two unknowns, cos(B) and cos(D). Solve those equations for cos(B) and cos(D) and then use the fact that cos^2(B)+ cos^2(D)= 1.
  • #1
gaobo9109
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Homework Statement


In an irregular quadrilateral ABCD, the length of all sides are AB=a BC=b CD=c DA=d and the length of the diagonal AC is x. Angle ABC + angle ADC = 180

prove that (ab+cd)x2=(ac+bd)(ad+bc)

Homework Equations



Cosine formula c2 = a2 + b2 - 2abcosθ

The Attempt at a Solution


I really have no idea how to start
 
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  • #2
Use the cosine formula, [itex]c^2= a^2+ b^2- 2abcos(C)[/itex], with x as "c", sides AB and BC as "a" and "b", and angle ABC as angle "C". Then do the same with x as "c" again but AD and BD as "a" and "b", and angle ADC as angle "C". Of course, the "x" in both of those is the same so they can be set equal.
 

Related to Finding the diagonal of an irregular quadrilateral

1. How do you find the diagonal of an irregular quadrilateral?

To find the diagonal of an irregular quadrilateral, you can use the Pythagorean Theorem. First, identify the two sides of the quadrilateral that intersect at a right angle. Then, use the formula c² = a² + b² to find the length of the diagonal.

2. Can you find the diagonal of an irregular quadrilateral if you only know the lengths of the sides?

Yes, it is possible to find the diagonal of an irregular quadrilateral if you know the lengths of the sides. You can use the formula c² = a² + b², where a and b are the lengths of two sides that intersect at a right angle. This will give you the length of the diagonal, also known as the hypotenuse in a right triangle.

3. Is there a specific formula for finding the diagonal of an irregular quadrilateral?

No, there is no specific formula for finding the diagonal of an irregular quadrilateral. However, you can use the Pythagorean Theorem or other geometric principles, such as the Law of Cosines, to calculate the length of the diagonal.

4. What if the quadrilateral is not a right angle, can I still find the diagonal?

Yes, you can still find the diagonal of an irregular quadrilateral even if it is not a right angle. In this case, you can use the Law of Cosines, which states that c² = a² + b² - 2ab cos(C), where C is the angle opposite the diagonal. This will give you the length of the diagonal.

5. Can the diagonal of an irregular quadrilateral be longer than any of its sides?

Yes, it is possible for the diagonal of an irregular quadrilateral to be longer than any of its sides. This can happen when the quadrilateral is very elongated or has a small angle between its sides, making the diagonal longer. It is also possible for the diagonal to be shorter than some of the sides, depending on the shape of the quadrilateral.

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