Solve for x and y

[SternBeast]

The task is to calculate x and y using measures OM, MP and angles α and β.
Can use trigonometric functions and Pythagoras.

Result should be: if OM = 5, MP = 3, α = 40◦ , β = 30◦, then x = 4.856 and y = 6.033
I tried using tan(α) = y/2 / x, but that didn't work.
Maybe I could connect points O and P and create a new triangle (although it isn't a right triangle)?

How to proceed with this triangle?

 

[Otis]

Hi SternBeast. We could break the shape into two right-triangles and a rectangle. Draw a horizontal line segment from point M to side y. (That segment will be the base of a right triangle having MP as its hypotenuse.) Draw a vertical line from point M to side x. (That segment will be the height of a right triangle having OM as its hypotenuse.) Using the right-triangle definitions for sine and cosine, you can find the lengths of each right-triangle's legs. Two of those lengths sum to x; the other two sum to y.

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[SternBeast]

Ok, thanks Otis! That helps a little.

 

[Otis]

… helps a little.

Glad to hear that. Feel free to let us know, if you'd like more guidance or have any questions. Please include any work that you've tried.

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[Merida]

Let the right angle be angle Q
Draw a line parallel to OQ from point M to touch at point say N
Using the fact that corresponding angles are equal , you will get PMN as 70 degrees
Now using sin alpha and sin PMN you can find y , similarly using cos alpha and cos PMN , you will get x