Question

ADA. 5B. 5.1C. 5.42D. 6.231213BFind the length of AD. Round the answer to the nearest hundredth.

Answer 1

Answer:
`AD\text{ = 5.42 \lparen option C\rparen}`

Step-by-step explanation:

Given:

DC = 13, BC = 12

To find:

the value of AD

To determine AD, we will apply the geometric mean formula:

`\begin{gathered} (leg)/(part)\text{ = }(hypotenuse)/(leg) \\ \\ We\text{ have 2 legs, for the part where there is AD} \\ AD\text{ = leg} \\ hypotenuse\text{ = AC} \\ part\text{ = AB} \\ The\text{ length of hypotenuse is unknown as BC was not given} \end{gathered}`
`\begin{gathered} We\text{ will calculate the 2nd leg to get the hypotenuse:} \\ 2nd\text{ }leg\text{ = DC = 13} \\ hypotenuse\text{ = AC } \\ AC\text{ = AB}+\text{ BC} \\ AC=\text{ AB + 12} \\ part\text{ for the 2nd leg = BC}=\text{ 12} \\ \\ Using\text{ the formula above:} \\ \frac{2nd\text{ leg}}{part\text{ for 2nd leg}}\text{ = }\frac{hyp}{2nd\text{ leg}} \\ \\ (13)/(12)\text{ = }\frac{AB\text{ + 12}}{13} \\ \\ 13(13)\text{ = 12\lparen AB + 12\rparen} \\ 169\text{ = 12AB + 144} \end{gathered}`
`\begin{gathered} 169\text{ - 144 = 12AB} \\ 25\text{ = 12AB} \\ AB=\text{ 25/12} \\ AB=\text{ 2.0833} \\ \\ AC\text{ = AB +}BC\text{ = 2.0833 + 12 = 14.0833} \\ hyp\text{ = AC}=\text{ 14.0833} \end{gathered}`

We can find AD since the hypotenuse is known using the formula:

`\begin{gathered} (AD)/(AB)\text{ = }(AC)/(AD) \\ \\ (AD)/(2.0833)\text{ = }(14.0833)/(AD) \\ \\ AD^2\text{ = 2.0833}*14.0833 \\ \\ AD^2=\text{ 29.3397} \end{gathered}`
`\begin{gathered} square\text{ root both sides:} \\ AD\text{ = }√(29.3397)\text{ = 5.4166} \\ \\ AD\text{ = 5.42 \lparen nearest hundredth\rparen \lparen option C\rparen} \end{gathered}`