Answer:
[tex]x\approx24^\circ[/tex]
Step-by-step explanation:
We know that r and s have a combined length of 70. Therefore:
[tex]r+s=70[/tex]
Notice that we can determine r by using the Pythagorean Theorem. In this case, r is the hypotenuse, and 20 and 8 are the legs. Therefore:
[tex]a^2+b^2=c^2[/tex]
Substituting 20 and 8 for a and b and r for c yields:
[tex]20^2+8^2=r^2[/tex]
Compute:
[tex]464=r^2[/tex]
Therefore:
[tex]r=\sqrt{464}=\sqrt{16\cdot 29}=4\sqrt{29}[/tex]
Now, we can determine s. We know that:
[tex]s+r=70[/tex]
So, by substitution:
[tex]s+4\sqrt{29}=70[/tex]
Therefore:
[tex]s=70-4\sqrt{29}[/tex]
Now, notice that, with respect to x, 20 is the opposite side and s is the hypotenuse.
Therefore, we can use the sine ratio. The sine ratio is the ratio between a right triangles opposite side to its hypotenuse.
In this case, the opposite to x is 20, and the hypotenuse is s, or 70-4√29. Therefore:
[tex]\displaystyle \sin(x^\circ)=\frac{20}{s}[/tex]
By substitution:
[tex]\displaystyle \sin(x^\circ)=\frac{20}{70-4\sqrt{29}}[/tex]
Take the inverse sine of both sides:
[tex]\displaystyle x^\circ=\sin^{-1}\Big(\frac{20}{70-4 \sqrt{29}} \Big)[/tex]
Use a calculator. Therefore:
[tex]x\approx24.37^\circ\approx24^\circ[/tex]
