Hey, this is Presh Talwalkar reminding you to Mind Your Decisions.
In triangle ABC, angle A is bisected into two equal angles of 60°.
If AD = 100, AB = 2(AC), then what it the length of BC?
This was asked in a math contest in Switzerland and no calculators were allowed.
Pause the video if you like to give it a try,
and when you are ready, keep watching to learn how to solve this problem.
So let's get started solving this problem.
If AC has a length of x, then AB = 2x.
Now we are going to prove that BD = 2 (DC).
This follows immediately by the angle bisector theorem,
but if you didn't remember the angle bisector theorem, let's prove it.
So let's get started by labeling ∠ADC as α.
Then ∠ADB would be suplementary,
so it would be 180° - α.
Next, we'll use the laws of sines,
which I read was first described by the Persian mathematician Tusi.
So let's use the laws of sines in △ADC.
This gives that DC / sin 60° = x /sin α.
Next we'll use the laws of sines in △ADB.
This gives that BD / sin 60° = 2x /sin (180° - α).
Next ,we use the identity that the sine of an angle is the same as its supplementary angle,
so we can simplify this fraction.
Now, let's multiply both equations by sin 60°,
and then we divide the first equation from the second equation.
We get the ratio BD / DC
We can now cancel out many terms,
we'll left with just the fraction equals to 2,
and this would directly leave to the identity that we want,
BD = 2 (DC).
So if DC has a length of y,
BD would have a length of 2y.
Now recall that the length of AD is 100.
Now we are going to use Al-Kashi's theorem.
y² = x² + 100² - 2 (x)(100) cos 60°
Since cos 60° = 1/2, we'll simplify this.
We'll put this equation to aside.
And now we'll use Al-Kashi's theorem on this triangle.
We again simplify cos 60° and we get this equation.
So we now have two equations.
Let's cancel out the y² terms,
we'll multiply the first equation by 4,
and then we would subtract the second equation.
We cancel terms between the equations
and left with 0 = 3(100²) - 200x,
we can then solve that x = 150.
We then go back to the diagram,
and then we are going to use the law of cosines, or Al-Kashi's theorem,
and we're going to substitute in for x.
We get that y² = 17500,
and then that means y, we wants to take the positive value because we're dealing with a length,
y would be equal to 50 times the square root of 7.
Now recall we want to solve for BC,
and BC = 3y, so we get 150 times the square root of 7,
and that's our answer.
This is a great trigonometry problem
you really have to use a lot of concepts
to work out the final answer,
and you can do it all without a calculator.
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