MHB How do I calculate the side of a rhombus using the bisector of an angle theorem?

  • Thread starter Thread starter Yankel
  • Start date Start date
  • Tags Tags
    Angle Theorem
Yankel
Messages
390
Reaction score
0
Hello all,

I have this question I struggle with...

View attachment 7836

EDFB is a parallelogram. It is known that AB/BC = AD/DC.

1) Prove that the parallelogram is a rhombus.

2) It is given that: AB=9, AC=10, BC=AD. Calculate the side of the rhombus.

I think I solved the first part. There is a theorem called the "bisector of an angle theorem" according to which if AB/BC = AD/DC then the line BD is a bisector of an angle of the angle B and then a parallelogram in which the diagonal is a bisector of an angle is a rhombus. Am I correct ?

I have a problem with the second part. I can't figure out how to solve it. The answer should be 3.6. I have tried the intercept theorem (or Thales' theorem), but couldn't figure it out.

Can you kindly assist to in the second part of the question ?

Thank you in advance !
 

Attachments

  • aaaa.PNG
    aaaa.PNG
    1.9 KB · Views: 106
Mathematics news on Phys.org
Yankel said:
Hello all,

I have this question I struggle with...
EDFB is a parallelogram. It is known that AB/BC = AD/DC.

1) Prove that the parallelogram is a rhombus.

2) It is given that: AB=9, AC=10, BC=AD. Calculate the side of the rhombus.

I think I solved the first part. There is a theorem called the "bisector of an angle theorem" according to which if AB/BC = AD/DC then the line BD is a bisector of an angle of the angle B and then a parallelogram in which the diagonal is a bisector of an angle is a rhombus. Am I correct ?
Yes.
Yankel said:
I have a problem with the second part. I can't figure out how to solve it. The answer should be 3.6. I have tried the intercept theorem (or Thales' theorem), but couldn't figure it out.

Can you kindly assist to in the second part of the question ?

If BC = AD = x, then the equation AB/BC = AD/DC becomes 9/x = x/(10-x), a quadratic for x with a unique positive solution.

From the similar triangles AED and ABC you can then calculate that ED/x = x/10, which gives ED = 3.6.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top