Math Destroyer Help

[torobcheh21]

I'm having trouble understanding this problem 38 from test 2 in math destroyer 2010:

Find the area of the isosceles triangle below if the two equal sides are of length 10 and meet at an angle 40.

The picture shows an isosceles triangle with a ten on each vertical slant length and a 40 degree symbol at the top of the vertex.

The answer is 100sin(20)cos(20). I don't understand how to get the sin 20 and cos 20 part. I understand that by drawing a perpendicular line down the middle you can make two right triangles each with an angle of 20 and a hypotenuse of 10. The answer says two identical right triangles are formed each of whose base and height is 10sin(20) and 10cos(20), respectively.

Does anyone know how they get this?

Thanks

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[NewBee12]

Yeah I remember this question.
the area of a triangle is: (1/2)bh.
the height is the perpendicular line. to find it use: cos(20)=h/10 then h = 10cos(20)
the base is twice the opposite side to the angle (20)
sin(20)=x/10 then x = 10 sin(20). the base is then 20 sin(20)
the area is then : (1/2) 20 sin(20) 10 cos(20) = 100sin(20)cos(20)

good luck

 

[torobcheh21]

thank you got it!

 

[torobcheh21]

Another problem I'm having a lot of trouble with comes from 2010 math destroyer test 3, #26...

A pleasure boat travels 240 miles downstream from A to B in 12 hours. If the speed of the stream is 4 mph, how long will it take the same boat to make the return trip upstream from B to A?

The answer is 20 hours, but I don't understand their calculation. I understand that the average mph is 20 and the total trip takes 12 hours, but when I tried to solve it, I added 4 mph for the trip there and subtracted 4 mph for the trip back. Giving me 24 mph for speed of the boat from A to B and 16 mph for the return trip. Can someone please take a look at the answer and explain to me why this logic is wrong.

Thank you

 

[NewBee12]

12 hours is only from A to B not for the round trip!!!
the overall speed from A to B is 240/12 = 20mph. since the speed of the river is 4mph the speed of the boat in still water is 16mph. for the return trip the overall speed is 16-4=12mph. Therefore t= 240/12 = 20 hours

 

[torobcheh21]

thanks newbee great explanation...careless mistake on my part. thanks!

 

[dl2501]

Yeah I remember this question.
the area of a triangle is: (1/2)bh.
the height is the perpendicular line. to find it use: cos(20)=h/10 then h = 10cos(20)
the base is twice the opposite side to the angle (20)
sin(20)=x/10 then x = 10 sin(20). the base is then 20 sin(20)
the area is then : (1/2) 20 sin(20) 10 cos(20) = 100sin(20)cos(20)

good luck

Old post I know, but in the answer choices for this, he said that the double angle for cosine is 50*sin(40)... I got the answer with the 100sin(20)cos(20), but how is the double angle derived?

Thank you!

@orgoman22