What is the range?

Geometry Level 3

In triangles $\triangle {ABC}$ and $\triangle {DEF}$ , $\angle A$ and $\angle D$ are equal. Sides $\overline {AB}$ and $\overline {DE}$ are of equal length. Sides $\overline {BC}$ and $\overline {EF}$ are also of equal length. Length of side $\overline {AC}$ is twice that of side $\overline {DF}$ . The ratio of lengths $\overline {BC}$ and $\overline {AB}$ must lie in the range $[p, q]$ . What is the magnitude of $3p+q$ ?

The answer is 2.

1 solution

Dan Czinege
May 12, 2019

So we can put those two triangles into one (as shown on the picture). And now we can set up equation by the rule of cosines for a:
$a^2=a^2+b^2/4-2*a*(b/2)cos(γ)$
$b=4acos(γ)$
and now for c:
$c^2=a^2+b^2/4-2*a*(b/2)cos(180°-γ)$ And here we can plug b=4acos(γ) and if we simplify that we will get this:
$c^2=a^2(1+8cos^2(γ))$
Now the ration is:
$a/c=sqrt(1/(1+8cos^2(γ)))$ And here we can easly see that minimum is when cos^2(γ)=1 and maximum when cos^2(γ)=0.
[p,q]=[1/3;1] So the answer is 2.

What is the range?

The answer is 2.

1 solution

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