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Answer:
x = (8/9)(4 +√7) ≈ 5.9073
Step-by-step explanation:
Maybe you want to know the value of x.
The Pythagorean theorem can be used to find the length of the bottom side of the right triangle. It is ...
bottom = √((5x)² -(3x)²) = √(16x²) = 4x
Then the cosine of the angle at lower left is ...
cos(α) = Adjacent/Hypotenuse
cos(α) = 4x/5x = 4/5
Now, the law of cosines can be used to find x. Using the angle whose cosine we just found, we have ...
(4x)² = (5x)² +8² -2(5x)(8)cos(α)
16x² = 25x² +64 -80x(4/5)
9x² -64x +64 = 0 . . . . . put in standard form
The quadratic formula can be used to find the solutions.
x = (-(-64) ±√((-64)² -4(9)(64)))/(2(9))
x = (64 ±√1792)/18 = (32 ±8√7)/9 ≈ {1.204, 5.907}
In this context, only the larger solution makes sense.
x = (8/9)(4 +√7) ≈ 5.9073
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Formulas used
Law of Cosines
For sides a, b, c and angle C opposite side c, ...
[tex]c^2=a^2+b^2-2ab\cdot\cos(C)[/tex]
Quadratic formula
For quadratic ax²+bx+c=0, the solutions are ...
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Pythagorean theorem
For legs a, b, and hypotenuse c of a right triangle, ...
c² = a² + b²
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Alternate solution
As an alternative to using the Law of Cosines, you can define "y" to be the length of the dashed bottom leg of the triangle whose hypotenuse is 4x. Two equations can be written using y and the Pythagorean theorem for the right triangles in which y is all or part of a leg. This set of equations can be solved for x to get the same result as the one shown above.
