Solve The Triangle

I am having a tough time trying to solve this problem.

I have utilized the 30, 60, 90 triangle measures for the length of sides. However, I am stuck since the side that would be √3 has 100 as its length. How do I solve then?

This is what I have done so far:

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Where you write $\sqrt 3$ in your working, you want to replace it with 100. – Haikal Yeo Jul 12 '14 at 15:02
If the length that you have marked as hypotenuse=y, then y=100/sin 30 and x=y/sin 60 – Vikram Jul 12 '14 at 15:08
Ok I just did that. I multiplied 100 (1/2) = 50. That can't be right. That would mean that one of the sides is larger than the hypotenuse? – user137452 Jul 12 '14 at 15:09
@user137452,100/(1/2)=200 – Vikram Jul 12 '14 at 15:14
You are looking at the hypotenuse all wrong, and so the miscalculations. hypotenuse is supposed to be the side opposite to the right angle, or $x$ in the original question. – MonK Jul 12 '14 at 16:28

Let's name the triangle as $\triangle ABC$ which is right angled at B so that $x$ is hypotenuse. And this is a 30-60-90 triangle.

Let the perpendicular from B, drop on the Hypotenuse at D giving $\triangle ABD$ and $\triangle BDC$, and both of these small triangle are 30-60-90 triangle. So, apply the trignometry identities.

The length $x$ will be $$=100\cdot tan 30^\circ+100 \cdot tan 60^\circ$$

Or, $$100 \cdot {\frac{4}{\sqrt 3}}$$

A little hint, remembering sin, cos and tan identities and values is lot simple and elegant than remembering thew cosec, sec and cot values.

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Stupid question but how did you recognize that the angle where both the triangles meet was 90 degrees? – user137452 Jul 12 '14 at 15:38
It is a perpendicular BD on AC :) – MonK Jul 12 '14 at 15:41
Would you mind drawning that for me? Thanks in advance. – user137452 Jul 12 '14 at 15:44
If only I knew how to draw in the answer, I would have already done it. Let me find out.. – MonK Jul 12 '14 at 15:45
Forget it. I see what you mean. – user137452 Jul 12 '14 at 15:45

To find $x$ here, you will need to look at this as 2 separate triangles, and find the base of each separate triangle, then add the bases up to get $x$.

For the triangle on the left hand side, we can use the $\tan(x)$ function to relate the $60^{\circ}$ angle with the opposite side of length 100, and our adjacent side, which we are looking for (let's call it $y$).

So, we know $\tan(x) = \dfrac{\text{opp}}{\text{adj}}$, and this gives $\tan(60^{\circ}) = \dfrac{100}{y}$. But $\tan(60^{\circ}) = \sqrt{3}$, so we have:

$\sqrt{3} = \dfrac{100}{y}$, and solving for $y$ gives $y = \dfrac{100}{\sqrt{3}}$.

Now, we need to solve for the adjacent side of the triangle on the right-hand side using the angle $30^{\circ}$. Again, we want to relate the opposite side of $30^{\circ}$ to its adjacent side (the unknown -- we can call it $z$). We will have to use $\tan(x)$ again. So we get:

$\tan(30^{\circ}) = \dfrac{\text{opp}}{\text{adj}} = \dfrac{100}{z}$. But, $\tan(30^{\circ}) = \dfrac{1}{\sqrt{3}}$, so we have the equation:

$\dfrac{1}{\sqrt{3}} = \dfrac{100}{z}$ and solving for $z$ gives:

$z = \dfrac{100}{\frac{1}{\sqrt{3}}} = 100\sqrt{3}$.

Finally, $x = y + z$, since $x$ is the length of both adjacent sides put together, so:

$x = \dfrac{100}{\sqrt{3}} + 100\sqrt{3} = \dfrac{100}{\sqrt{3}} + \dfrac{100\sqrt{3}}{1}*\dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{100 + 100*3}{\sqrt{3}} = \dfrac{400}{\sqrt{3}}$, which is your final answer.

Here is the picture to help you see what I am doing:

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So if we were to rationalize the denominator the answer would be 400√3 /3? – user137452 Jul 12 '14 at 15:28
Ok, I got it. Thanks for the explanation. – user137452 Jul 12 '14 at 15:31
Yes, you are right, if you decide to rationalize the denominator (some teachers require that you do, others don't), then the answer is $\dfrac{400\sqrt{3}}{3}$. – user46944 Jul 12 '14 at 15:46

First, lets take a generic hypotenuse-opposite-adjacent right triangle and divide each side by the length of the opposite side (opposite to angle $t$). This gives us a similar triangle with sides $\csc t$, $1$, and $\cot t$. Next, multiply each side by 100 (see below) to get another similar triangle with a height of $100$, hypotenuse of $100\csc t$, and a horizontal base of $100\cot t$.

Now we want to apply our result to a $60$ degreed, and a $30$ degreed right triangle, and add the horizontal bases together.

Our result becomes $$100\cot 60^\circ + 100\cot 30^\circ=100(\sqrt{3}+1/\sqrt{3})=400/\sqrt{3}$$

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Both of the current answers utilize trig functions. I thought I would add an answer that doesn't use trig functions, but rather uses scaling of two $(30,60,90)$ triangles.

We are used to seeing the sides of a $(30, 60, 90)$ triangle being written as $(1, \sqrt{3}, 2)$ (where $1$ is the length opposite to the $30$ degree angle, $\sqrt 3$ is the length opposite to the $60$ degree angle, etc). We can then scale the side lengths by whatever multiplier we wish. The diagram below shows how to scale the side lengths appropriately for this problem.

For the left-hand triangle (top), we choose the multiplier $\frac{100}{\sqrt 3}$ (because the side opposite the $60$ degree angle is $100$ units long). For the right-hand triangle (bottom), we choose the multiplier $100$ (because the side opposite the $30$ degree angle is $100$ units long).

Thus, the length $x$ can be found easily: $$x = 100\sqrt{3} + \frac{100}{\sqrt 3} = \frac{400\sqrt{3}}{3}$$

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