Question

what is the value of ( y )? ( 3sqrt{3} ) units ( 6sqrt{3} ) units ( 9sqrt{3} ) units ( 12sqrt{3} ) units

Explanation:

Step1: Find length of NM

$NM = NU + UM = 9 + 3 = 12$

Step2: Find length of TM

$TM = 6$

Step3: Use geometric mean theorem

For right triangle $NTM$ with altitude $TU$, $\frac{y}{NM} = \frac{NU}{y}$
Rearrange to $y^2 = NM \times NU$

Step4: Substitute values and solve

$y^2 = 12 \times 9 = 108$
$y = \sqrt{108} = 6\sqrt{3}$

Answer:

$6\sqrt{3}$ units