+349 In triangle $STU$, let $M$ be the midpoint of $\overline{ST},$ and let $N$ be on $\overline{TU}$ such that $\overline{SN}$ is an altitude of triangle $STU$. If $ST = 13$, $SU = 14$, $TU = 4$, and $\overline{SN}$ and $\overline{UM}$ intersect at $X$, then what is $UX$?
+15046
What is UX?
142−x2=42−(x−13)2196−x2=16−x2+26x−16926x=349xU=13.423yU=√196−13.4232yU=3,978xM=6.5yM=0
fUT(x)=yU−yTxU−xT⋅(x−13)=3.978−013.432−13⋅(x−13)fUT(x)=9.2083x−119.7079fUM(x)=yU−yMxU−xM⋅(x−6.5)=3.978−013.432−6.5⋅(x−6.5)fUM(x)=0.5739x−3.73035mUT=9.2083mSN=−1mUT=−19,2083mSN=−0.1086fSN(x)=−0.1086x
check!
The point N is not on the section ¯UT. Still, carry on!
fUX(x)=fUM(x)=0.5739x−3.730350.5739x−3.73035=−0.1086x0.6825x=3.73035xX=5.4657yX=0.5739x−3.73035=0.5739⋅5.4657−3.73035yX=−0.59358
¯UX=√(yU−yX)2+(xU−xX)2¯UX=√(3.978+0.594)2+(13.423−5.466)2¯UX=9.177
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