Question

1. in the diagram below of right triangle kmi, altitude ig is drawn to hypotenuse km. if kg = 9 and ig = 12, the length of im is

Explanation:

Step1: Use the geometric mean theorem (altitude-on-hypotenuse theorem)

In a right - triangle, if an altitude \(IG\) is drawn to the hypotenuse \(KM\), then \(IG^{2}=KG\times GM\). Given \(KG = 9\) and \(IG = 12\), we substitute these values into the formula: \(12^{2}=9\times GM\).

Step2: Solve for \(GM\)

From \(144 = 9\times GM\), we can find \(GM=\frac{144}{9}=16\).

Step3: Use the geometric mean theorem for another relation

Also, in right - triangle \(KMI\), \(IM^{2}=GM\times KM\). Since \(KM=KG + GM=9 + 16=25\) and \(GM = 16\), then \(IM^{2}=16\times25\).

Step4: Calculate \(IM\)

Take the square root of \(IM^{2}\). \(IM=\sqrt{16\times25}=\sqrt{16}\times\sqrt{25}=4\times5 = 20\).

Answer:

\(20\)