Question

1. all the angles around point l in the shape below are right angles. how long is segment sm? leave answers in square root format (dont round).

Explanation:

Step1: Use the property of similar triangles

Since all angles around point \(L\) are right - angles, \(\triangle ULE\sim\triangle SLM\). The ratio of corresponding sides of similar triangles is equal. We know that \(\frac{UL}{SL}=\frac{EL}{ML}\). Also, by the property of the ratio of areas or the ratio of side - lengths in similar right - angled configurations (or using the concept of the product of segments in intersecting lines in a right - angled 'cross' - like structure).
Another approach is to use the fact that in right - angled configurations with intersecting lines, we can apply the formula based on the similarity of triangles formed. Let's use the formula \(UL\times LM = EL\times SL\). We know \(UL = 3\mathrm{cm}\), \(EL=20\mathrm{cm}\), \(SL = 15\mathrm{cm}\). First, we find \(LM\) from \(\frac{UL}{SL}=\frac{EL}{ML}\), so \(ML=\frac{EL\times SL}{UL}=\frac{20\times15}{3}=100\mathrm{cm}\).

Step2: Use the Pythagorean theorem in \(\triangle SLM\)

In right - triangle \(\triangle SLM\), by the Pythagorean theorem \(SM=\sqrt{SL^{2}+LM^{2}}\). Substitute \(SL = 15\mathrm{cm}\) and \(LM = 100\mathrm{cm}\) into the formula. \(SM=\sqrt{15^{2}+100^{2}}=\sqrt{225 + 10000}=\sqrt{10225}=5\sqrt{409}\mathrm{cm}\)

Answer:

\(SM = 5\sqrt{409}\mathrm{cm}\)