Question

in the following triangle, find length pm.
round to the nearest tenth.
pm =

Explanation:

Step1: Use the property of similar triangles

Since \( LO\parallel MN\), \(\triangle LOP\sim\triangle MNP\). The ratio of corresponding sides of similar triangles is equal. Let \(PM = x\). Then \(\frac{LO}{MN}=\frac{LP}{PM}\). Here \(LO\) is related to \(11.9\), \(MN = 17\), \(LP=13.8\), and \(PM=x\).
By the basic proportionality theorem (Thales' theorem), we can also use the proportion \(\frac{11.9}{17}=\frac{13.8}{13.8 + x}\) (because \(LP + PM=LM\)).
Cross - multiply: \(11.9\times(13.8 + x)=17\times13.8\).
Expand: \(11.9\times13.8+11.9x=17\times13.8\).
\(164.22+11.9x = 234.6\).
Subtract \(164.22\) from both sides: \(11.9x=234.6 - 164.22\).
\(11.9x=70.38\).
Solve for \(x\): \(x=\frac{70.38}{11.9}\approx5.9\).

Answer:

\(5.9\)