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Radda [10]
3 years ago
14

98 POINTS MATH!

Mathematics
1 answer:
patriot [66]3 years ago
6 0

Let a be the length of the leg with one tick mark and b the length of the leg with two tick marks.

In the upper triangle, the law of cosines says

6^2=a^2+b^2-2ab\cos43^\circ

In the lower triangle, it says

4^2=a^2+b^2-2ab\cos(4y-5)^\circ

Subtract the second equation from the first to eliminate a^2+b^2:

6^2-4^2=-2ab(\cos43^\circ-\cos(4y-5)^\circ)

10=ab(\cos(4y-5)^\circ-\cos43^\circ)

a and b are lengths so they must both be positive. 10 is also positive, so in order to preserve the sign on both sides of this equation, we must have

\cos(4y-5)^\circ-\cos43^\circ>0

\cos(4y-5)^\circ>\cos43^\circ

Now we have to be a bit careful. If x is an acute angle, then as x gets larger, the value of \cos x gets smaller. So if we have two angles \theta and \varphi, with 0^\circ, then we would have \cos\theta>\cos\varphi.

This means in our inequality, taking the inverse cosine of both sides would reverse the inequality:

(4y-5)^\circ

We know that (4y-5)^\circ is an angle in a triangle, so it must be some positive measure:

(4y-5)^\circ>0^\circ\implies y>\dfrac54^\circ

So we must have

\dfrac54^\circ

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