finding a leg length\nwhat is the length of side ts?\n2√6 units\n6√6 units\n24 units\n8 units

Answer

Explanation:

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

In a right - triangle, if an altitude is drawn to the hypotenuse, then the length of each leg of the right - triangle is the geometric mean of the length of the hypotenuse and the segment of the hypotenuse adjacent to that leg. For right - triangle (QST) with altitude (TR), we have the formula (TS^{2}=SR\times SQ). Given (SR = 12) and (SQ=6 + 12=18).

Step2: Substitute the values into the formula

Substitute (SR = 12) and (SQ = 18) into (TS^{2}=SR\times SQ). Then (TS^{2}=12\times18). Calculate (12\times18=216).

Step3: Find the value of (TS)

Take the square root of both sides: (TS=\sqrt{216}). Simplify (\sqrt{216}=\sqrt{36\times6}=6\sqrt{6}).

Answer:

(6\sqrt{6}\text{ units})