Question

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.

Answer 1

`BN=4\ \text{in}`

`AN=2\ \text{in}`

`AC=√(5)\ \text{in}`

Explanation:

Let
`BN=x`

`BC=c`

`NC=y=1\ \text{in}`

`AB=a=2√(5)\ \text{in}`

`AC=b`

We have the relation

`(BC)/(AB)=(AB)/(BN)\\\Rightarrow (c)/(a)=(a)/(x)\\\Rightarrow c=(a^2)/(x)\\\Rightarrow x+1=(a^2)/(x)\\\Rightarrow x^2+x=20\\\Rightarrow x^2+x-20=0\\\Rightarrow x=(-1\pm √(1^2-4* 1* \left(-20\right)))/(2* 1)\\\Rightarrow x=4,-5`

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

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

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

`b=√(h^2+y^2)\\\Rightarrow b=√(2^2+1^2)\\\Rightarrow b=√(5)\ \text{in}`

`\boldsymbol{AC=√(5)\ \text{in}}`