Question

How do I find the area in square units? What is the length of AC?B107.5A6.5DC

Answer 1

Given: A triangle ABC with altitude BD=6.5 units and side AB=7.5 units, BC=10 units.

Required: To determine the length of AC.

Explanation: The triangle ABD and triangle BCD are right-angled triangles. Hence we can apply Pythagoras theorem which states that

$(Hypotenuse)^2=(Perpendicular)^2+(Base)^2$

Hence for triangle ABD, we can write

$\begin{gathered} AB^2=BD^2+AD^2 \\ (7.5)^2=(6.5)^2+AD^2 \end{gathered}$

or,

$\begin{gathered} AD=\sqrt{(7.5-6.5)(7.5+6.5)} \\ AD=\sqrt{14}\text{ units} \\ AD=3.74\text{ units} \end{gathered}$

Similarly, for triangle BCD, we have

$\begin{gathered} 10^2=6.5^2+CD^2 \\ CD=\sqrt{(10+6.5)(10-6.5)} \\ CD=\sqrt{16.5\times3.5} \\ CD=\sqrt{57.75} \\ CD=7.599\text{ units} \end{gathered}$

Now,

$\begin{gathered} AC=AD+CD \\ =3.74+7.599 \\ =11.34\text{ units} \end{gathered}$

Final Answer: The length of AC is 11.34 units.