MHB Find All Possible Values of $AD$ in Cyclic Quadrilateral

  • Thread starter Thread starter anemone
  • Start date Start date
  • Tags Tags
    Cyclic
AI Thread Summary
In the cyclic quadrilateral $ABCD$ with equal sides $AB=BC=CA$, the diagonals $AC$ and $BD$ intersect at point $E$. Given the lengths $BE=19$ and $ED=6$, the problem requires finding all possible values of $AD$. Using the properties of cyclic quadrilaterals and the intersecting chords theorem, the relationship between the segments can be established. The calculations lead to potential values for $AD$, which can be derived from the known lengths and geometric properties. The solution ultimately reveals the possible lengths for side $AD$ in this configuration.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
$ABCD$ is a cyclic quadrilateral such that $AB=BC=CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE=19$ and $ED=6$, find all the possible values of $AD$.
 
Mathematics news on Phys.org
[TIKZ]
\draw circle (4) ;
\coordinate [label=left:$A$] (A) at (210:4) ;
\coordinate [label=above:$B$] (B) at (90:4) ;
\coordinate [label=right:$C$] (C) at (330:4) ;
\coordinate [label=below:$D$] (D) at (290:4) ;
\coordinate [label=above right:$E$] (E) at (intersection of A--C and B--D) ;
\draw (A) -- (B) -- node[ right ]{$x$} (C) -- (D) -- (A) --(C) ;
\draw (B) -- node[ right ]{$19$} (E) -- node[ right ]{$6$} (D) ;
\draw (-0.1,3.3) node{$\theta$} ;[/TIKZ]
Let $x$ be the side length of the equilateral triangle $ABC$, and $\theta$ the angle $ABD$, as in the diagram.

By the sine rule in triangle $ABE$, $\dfrac{19}{\sin 60^\circ} = \dfrac x{\sin(\theta+60^\circ)}$.

By the sine rule in triangle $ABD$, $\dfrac{x}{\sin 60^\circ} = \dfrac {25}{\sin(\theta+60^\circ)} = \dfrac{AD}{\sin\theta}$.

Therefore $\dfrac{19}x = \dfrac x{25}$ and hence $x = 5\sqrt{19}$. Then $\sin(\theta+60^\circ) = \dfrac{25\sin60^\circ}x = \dfrac {5\sqrt3}{2\sqrt{19}}$ and $\sin^2(\theta+60^\circ) = \dfrac{75}{76}$. So $\cos^2(\theta+60^\circ) = \dfrac{1}{76}$ and $\cos(\theta+60^\circ) = \pm\dfrac1{2\sqrt{19}}.$ It follows that $$\begin{aligned}\sin\theta = \sin((\theta+60^\circ) - 60^\circ) &= \sin(\theta+60^\circ)\cos60^\circ - \cos(\theta+60^\circ)\sin60^\circ \\ &= \frac{5\sqrt3}{2\sqrt{19}}\cdot\frac12 \pm \frac1{2\sqrt{19}}\cdot\frac{\sqrt3}2 \\ &= \frac{\sqrt3}{\sqrt{19}} \text{ or } \frac{3\sqrt3}{2\sqrt{19}}.\end{aligned}$$ Then $x\sin\theta = 5\sqrt3$ or $\dfrac{15}2\sqrt3$, so from the above sine rule $AD = \dfrac{x\sin\theta}{\sin60^\circ} = 10$ or $15$.

The above diagram shows the longer alternative $AD = 15$, with $CD = 10$. The other alternative comes from interchanging $A$ and $C$.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
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...
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...
Back
Top