Maximize the largest side

Geometry Level 4

In a triangle $ABC$ , $a = 7$ and $\sin A = \dfrac{11}{13}$ where $0<A<\dfrac{\pi}{2}.$ Given these details find the length of the largest possible side this triangle can have.

If this length is of the form $\dfrac{x}{y}$ , where $x$ and $y$ are coprime positive integers, find $x+y.$

Inspiration .

The answer is 102.

2 solutions

Md Zuhair
Jan 20, 2017

Relevant wiki: Sine Rule (Law of Sines)

Let $b$ be the largest side with opposite angle to be $B^o$

By Sine Rule, $\frac{a}{\sin A}$ = $\frac{b}{\sin B}$

hence Putting values of a and sin A we get ,

$b = \sin B$ x $\frac{91}{11}$

Now Max value of $\sin B = 1$

Hence for $b_{max}$ we need $\sin B = 1$ . Hence $b = \frac{91}{11}$

You should use \sin rather than sin in the LaTeX. I fixed it for you this time.

Jason Dyer Staff - 4 years, 4 months ago

Ok. I will remember it.

Md Zuhair - 4 years, 4 months ago

Akeel Howell
Jan 20, 2017

We are given that $a=7$ and $\sin A=\dfrac{11}{13}$ so considering the possibility of triangle $ABC$ being a right triangle, we see that $\dfrac{11}{13}=\dfrac{7}{h},$ because $\sin A = \dfrac{\text{Opposite(A)}}{\text{Hypotenuse}},$ where $h$ is the length of the hypotenuse of such a triangle. This leaves us with $\dfrac{11}{91}=\dfrac{1}{h} \\ \implies h=\dfrac{91}{11} \text{ and } h>7. \\ \therefore x=91, y=11 \Longrightarrow x+y=91+11=102$

You did not give an evidence that h is the largest length only valid for a right triangle! How we know the side length will increase for an obtused angled triangle!

Surely this is correct resulting by numerical analysis. ;-)

Andreas Wendler - 4 years, 4 months ago

We are given angle $A$ and the side opposite to it so if we were to increase the magnitude of the right angle and keep angle $A$ the same, then the length of the side that was the hypotenuse in the right triangle would increase. However, we would no longer have a triangle unless we were to extend side $a$ as well to make it longer. The problem with this is that side $a$ has a fixed length of $7.$ There would be a similar effect if we were to decrease the magnitude of the right angle, except that triangles would exist, but their largest possible side would not be as large as possible.

Akeel Howell - 4 years, 4 months ago

No evidence!

Andreas Wendler - 4 years, 4 months ago

@Andreas Wendler – The proof of this comes from the Sine Rule. $\dfrac{a}{\sin{A}}=\dfrac{b}{\sin{B}} \implies b = \sin{B} \cdot \dfrac{a}{\sin{A}}.$ The largest of $b$ therefore occurs when $\sin{B}=1.$ This happens to be when the triangle is a right triangle.

Akeel Howell - 4 years, 4 months ago

Maximize the largest side

The answer is 102.

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