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emmasim [6.3K]
3 years ago
8

In the right ΔABC, AN is the altitude to the hypotenuse. Find BN, AN, and AC, if AB=2√5 in and NC= 1 in.

Mathematics
1 answer:
kipiarov [429]3 years ago
8 0

Answer:

BN=4\ \text{in}

AN=2\ \text{in}

AC=\sqrt{5}\ \text{in}

Step-by-step explanation:

Let BN=x

BC=c

NC=y=1\ \text{in}

AB=a=2\sqrt{5}\ \text{in}

AC=b

We have the relation

\dfrac{BC}{AB}=\dfrac{AB}{BN}\\\Rightarrow \dfrac{c}{a}=\dfrac{a}{x}\\\Rightarrow c=\dfrac{a^2}{x}\\\Rightarrow x+1=\dfrac{a^2}{x}\\\Rightarrow x^2+x=20\\\Rightarrow x^2+x-20=0\\\Rightarrow x=\frac{-1\pm \sqrt{1^2-4\times 1\times \left(-20\right)}}{2\times 1}\\\Rightarrow x=4,-5

\boldsymbol{BN=4\ \text{in}}

h=\sqrt{a^2-x^2}\\\Rightarrow h=\sqrt{(2\sqrt{5})^2-4^2}\\\Rightarrow h=2\ \text{in}

\boldsymbol{AN=2\ \text{in}}

b=\sqrt{h^2+y^2}\\\Rightarrow b=\sqrt{2^2+1^2}\\\Rightarrow b=\sqrt{5}\ \text{in}

\boldsymbol{AC=\sqrt{5}\ \text{in}}

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