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the ratio of interior angles of a triangle is1:5:6. the shortest side is 4m. what is the area of the triangle?

Sagot :

pathogenic
1:5:6 ration of a triangle will give you a set of angles which is 15 degrees, 75 degrees and 90 degrees -- which means that you have right triangle.
Being the smallest angle, 15 degrees is the side that is opposite to the 4m.
Making use of the trigonometric functions (the SOH-CAH-TOA thingy), you'll need to get the side that is opposite to the 75 degree-angle for the calculation of area.

To get the other leg:
Sine 15 degrees = 4m/ Hypotenuse, then you will just derive the formula of hypotenuse here and it will be Hypotenuse = 4m/sine 15 degrees
Cosine 15 degrees = Adjacent/ hypotenuse, then it will be hypotenuse = adjacent/cosine 15 degrees, but we already know the equivalent of hypotenuse so we will use it as (4m/sine 15 degrees = adjacent/cosine 15 degrees). Deriving the measure of the adjacent side, we will have 4(cosine 15 degrees)/ sine 15 degrees.
Which is equal to:
[tex] \frac{4(1+ \sqrt{3} )}{\sqrt{3} -1} [/tex]

To get the area, we would just multiply the adjacent side to the 2m (because we already divided it by 2 so we would not need to after).

And we will arrive to [tex] \frac{8 + 8 \sqrt{3} }{ \sqrt{3} - 1} [/tex] , however simplifying it could get us to [tex] 16 + 8 \sqrt{3} [/tex] or approximation of 29.856 square meters.
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