Question

Given right triangle ABC with altitude BD drawn to hypotenuse AC. If AB = 6 and AD = 2, what is the length of AC? (Note: the figure is not drawn to scale.) B 6 2 D Answer: Submit Answer

Answer 1

The first step is to make a sketch of the triangle

The altitude (h= BD) of the triangle divides it into two similar right triangles and the hypothenuse, AC, into two line segments n= AD and m= DC.

The relationship between the altitude and the parts of the hypothenuse follows the ratios:

`(n)/(h)=(h)/(m)`

So, the first step is to determine the altitude of the triangle. To do so, you have to work with ΔABD, "h" is one of the sides of the triangle. Using the Pythagorean theorem you can determine the measure of the missing side:

`a^2+b^2=c^2`

Write the expression for the missing side:

`\begin{gathered} b^2=c^2-a^2 \\ \sqrt[]{b^2}=\sqrt[]{c^2-a^2} \\ b=\sqrt[]{c^2-a^2} \end{gathered}`

Replace c=6 and a=2

`\begin{gathered} h=\sqrt[]{6^2-2^2} \\ h=\sqrt[]{36-4} \\ h=\sqrt[]{32} \\ h=4\sqrt[]{2} \end{gathered}`

Now that we have determined the value of the altitude, we can calculate the value of m

`(n)/(h)=(h)/(m)`

Write the expression for m:

-Multiply both sides by m to take it from the denominators place:

`\begin{gathered} m\cdot(n)/(h)=m\cdot(h)/(m) \\ m\cdot(n)/(h)=h \end{gathered}`

-Multiply both sides of the equal sign by the reciprocal of n/h

`\begin{gathered} m((n)/(h)\cdot(h)/(n))=h\cdot(h)/(n) \\ m=(h\cdot h)/(n) \\ m=(h^2)/(n) \end{gathered}`

Replace the expression with h=4√2 and n=2 and calculate the value of m

`\begin{gathered} m=(h^2)/(n) \\ m=\frac{(4\sqrt[]{2})^2}{2} \\ m=(32)/(2) \\ m=16 \end{gathered}`

So DC=m= 16cm and AD=n= 2cm, now you can determine the measure of the hypothenuse:

`\begin{gathered} AC=AD+DC \\ AC=2+16 \\ AC=18 \end{gathered}`

The hypothenuse is AC=18cm