SOLUTION: Find the value of x, correct to 1 decimal place. http://imgur.com/a/WEA2Q Please show all working. Thanks in advance!

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Question 1045632: Find the value of x, correct to 1 decimal place.
http://imgur.com/a/WEA2Q
Please show all working. Thanks in advance!
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39623) (Show Source):
You can put this solution on YOUR website!
Look at and analyze the obtuse triangle first.

The obtuse angle next to the 50 degree angle is 130 degrees. The acute angle below is some value

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This obtuse triangle having TWO equal angles is therefore an isosceles triangle. The slanted segment is therefore 10 units, as well as the labeled segment at top being 10 units.

Now examine the right triangle on the left in the figure.
, in reference to the 50 degree angle, hypotenuse is that slanted segment of 10 units, and "x" is the opposite side from the 50 degree angle.
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Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
let y - the segment on the horizontal segment of the triangle that gets added to 10 to make up the whole length of the horizontal segment.

this forms 2 right triangles
.

the tangent of 25 degrees is equal to x / (y + 10) in the longer right triangle.

the tangent of 50 degrees is equal to x / y) in the shorter right triangle.

you can solve for y and then solve for x, or you can solve for x directly.

we'll solve for y and then solve for x since it looks like it may be easier than solving for x directly.

i did it both ways and got the same answer.

start with:

tan(25) = x / (y + 10) and tan(50) = x/y

solve for x in each equation.

start with tan(25) = x / (y + 10)
multiply both sides of that equation by (y + 10) to get:
tan(25) * (y + 10) = x

start with tan(50) = x/y
multiply both sides of that equation by y to get:
tan(50) * y = x

both expressions are equal to x, so both expressions must be equal to each other.

you get tan(25) * (y + 10) = tan(50) * y

simplify to get tan(25) * y + tan(25) * 10 = tan(50) * y

subtract tan(25) * y from both sides of the equation to get:

tan(25) * 10 = tan(50) * y - tan(25) * y

factor the right side of the equation to get:

tan(25) * 10 = y * (tan(50) - tan(25))

divide both sides of the equation by (tan(50) - tan(25)) to get:

tan(25) * 10 / (tan(50) - tan(25)) = y

solve for y to get y = 6.427876097

since you know that tan(50) = x/y, then solve for x to get x = y * tan(50).

this results in x = 7.660444431

round it to 1 decimal place and your solution is that x = 7.7