Question

a picture of el castillo pyramid is shown with \\( \triangle cms \\) drawn over it. points v, h, and g are midpoints of each side of the triangle. the medians \\( \overline{gm} \\), \\( \overline{ch} \\), and \\( \overline{sv} \\) are shown. \\( mk=(8x + 2) \\) meters (m), and \\( kg=(\frac{4}{3}x + 7) \\) m. if kyra is standing at point k, how far did she walk up the stairs from point g?

Explanation:

Step1: Use the property of the centroid of a triangle

The centroid of a triangle divides each median in a ratio of \(2:1\). That is, \(MK = 2KG\).
Given \(MK=(8x + 2)\) and \(KG=(\frac{4}{3}x + 7)\), we set up the equation \(8x+2 = 2(\frac{4}{3}x + 7)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(8x+2=\frac{8}{3}x + 14\).
Subtract \(\frac{8}{3}x\) from both sides: \(8x-\frac{8}{3}x+2=\frac{8}{3}x-\frac{8}{3}x + 14\).
\(\frac{24x - 8x}{3}+2=14\), \(\frac{16x}{3}+2 = 14\).
Subtract 2 from both sides: \(\frac{16x}{3}+2-2=14 - 2\), \(\frac{16x}{3}=12\).
Multiply both sides by \(\frac{3}{16}\): \(x=\frac{12\times3}{16}=\frac{9}{4}\).

Step3: Find the length of \(KG\)

Substitute \(x = \frac{9}{4}\) into the formula for \(KG\).
\(KG=\frac{4}{3}x+7=\frac{4}{3}\times\frac{9}{4}+7\).
\(KG = 3 + 7=10\).

Answer:

\(10\) meters.