Question

Please help due tonight

Answer 1

x = 24, y = 12

Explanation:

We can use the law of sines to solve for x

The law of sines says the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides (and angles)

In this triangle

the side with length 12√3 is opposite 60°

the side with length x is opposite 90°

Using the law of sines,

`(12√(3))/(\sin60)=(x)/(\sin 90)\\`

sin 60 =
`(√(3))/(2)`

sin 90 = 1

`\rightarrow (12√(3))/((√(3))/(2))=(x)/(1)\\\\\\x = 12√(3) / (√(3))/(2)}\\\\\textrm{When dividing fractions, flip the divisor and multiply}\\\\\rightarrow 12√(3) * (2)/(√(3))\\\\\textrm{$√(3)$ cancels out from numerator and denominator}:\\\\\\\rightarrow x = 12 * 2 = \boxed{24}\\\\\\`

To solve for y we can again use the law of sines

`(y)/(\sin 30) = (x)/(\sin 90)\\\\\textrm{again, as before $\sin 90$ is 1}\\\\\rightarrow (y)/(\sin 30) = x\\\\\rightarrow x = (y)/(\sin 30)\\\\\rightarrow y = x \sin 30\\\\\sin30 = (1)/(2)`

So y = x/2 = 24/ 2 = 12

ans:
x = 24, y = 12

Note
Another way to solve this question is to first recognize the triangle as a 30-60-90 right triangle

In a triangle of this kind, if the longest side side is a, the shortest side is a/2, and the third side is √3/2a

We could have solved this in this manner also

Here longest side = x, shortest side is y

The other side is 12√3 = x √3/2

So x = 12√3 ÷ √3/2 = 24 (as in the first method)

y = x/2 = 12

This approach is ok too, I find it less intuitive and we have to remember the ratios