dierdurst

You can first find the second \(a\), by setting up an equation and solving it. We can make the equation \(\frac{2}{5} = \frac{a}{3}\), which we can solve using cross-multiplication. Then, we get \(6 = 5a\), which gives us \(a = \frac{6}{5}\). So if \(a = \frac{6}{5}\), and \(a\) is inversely proportional to \(c\), than we can see that \(\frac{6}{5} \times c\) must equal \(18\), because \(2 \times 9 = 18\). \(\frac{18}{\frac{6}{5}} = 18 \times \frac{5}{6} = 15\). So \(c = 15\).

- Daisy

Nevermind, I have figured it out!

- Daisy

Haha, no problem!

- Daisy

We have that angle \(ABC = 49\), and that \(ACB = 12\), so we can see that \(BAC = 119\). \(FHE = BHE\) and we can find \(FHE\) by calculating the angle degrees in \(FHEA\), by doing \(360 - 90(AFH) - 90(AEH) - 119(FAE) = 61 = FHE = BHC\), so your answer would be \(61\).

- Daisy

That makes sense, too! Thanks Chris!

- Daisy

We can use case work counting with this. We have three cases, if we start with \(1\) step, \(2\) steps, or \(3\) steps. You then can add a subcase of \(1\) step, \(2\) steps, or \(3\) steps to the beginning cases. You can keep doing this until all your cases add up to \(9\). This is a hassle, but it works. I don't reccommend this method, but I do not know any other methods, sorry. The end answer is \(149\). Haha I hope this helped! Hopefully you can find a better solution!

- Daisy

Inversely proportional means basically that the product of the numerator and denominator will always be the same. So if the fraction is \(\frac{16}{21} = \frac{j}{k}\), the product of \(16\times21 = 336\). We are trying to find \(j\) if \(k = 14\). We have \(\frac{16}{21} = \frac{j}{14}\). You can solve this by doing \(336\) divided by \(14\), which gives you \(24\). \(j = 24\).

- Daisy

\(\Delta DEB\)  is a \(30 - 60 - 90\) triangle, so \(BE = \sqrt{3}(DE)\). \(EC = DE\), and since we are only looking for ratios, we can assign \(DE\) and \(EC\) to \(x\). \(\Delta ABC\) is a \(30 - 60 - 90\) triangle also. \(AC\) is \(4x\) because \(BD\) is a median, therefore splitting \(AC\) into two. Since \(AC = 4x\), \(BC = 2x\). (If you didn't know already, a \(30 - 60 - 90\) triangle's sides are in a ratio of \(1 : \sqrt{3} : 2\).) If \(BC = 2x\), then \(AB = 2\sqrt{3}x\), and \(EC = x\), so your ratio would be \(\frac{2\sqrt{3}x}{x}\), which simplifies to \(2\sqrt{3}\) when you cancel the \(x\)'s.

- Daisy

If we prime factorize \(169\), we can get \({13}^{2}\), which makes \(169\) a perfect square. Therefore, \(\sqrt{169} = 13\)

- Daisy

If we prime factorize  \(900\), we get \({2}^{2} \times {3}^{2} \times {5}^{2}\). All of the factors are squares, so we can see that 900 must be a perfect square. If we multiply \(2 \times 3 \times 5\), we get \(30\), which is the answer.

- Daisy