Answer: (D) P=24√6, A=96√3
Step-by-step explanation:
Consider ΔABC where D is the midpoint of BC. Since ABC is an equilateral triangle, then segment AD is a perpendicular bisector with length of 12√2. This creates ΔADC which is a 30°-60°-90° triangle.
Now you can use the rules for this special triangle to find the length of the hypotenuse.
30° ⇄ side length "a" base - DC on ΔADC
60° ⇄ side length "a√3" height - AD on ΔADC
90° ⇄ side length "2a" hypotenuse - AC on ΔADC
Step 1: solve for "a"
[tex]AD: a\sqrt3=12\sqrt{12}[/tex]
[tex]\dfrac{a\sqrt3}{\sqrt3}=\dfrac{12\sqrt2}{\sqrt3}[/tex]
[tex]a=\dfrac{12\sqrt2}{\sqrt3}\bigg(\dfrac{\sqrt{3}}{\sqrt{3}}\bigg)[/tex]
[tex]= \dfrac{12\sqrt6}{3}[/tex]
[tex]=4\sqrt6[/tex]
Step 2: solve for "2a"
[tex]AC: 2a =2(4\sqrt{6})[/tex]
[tex]=8\sqrt{6}[/tex]
Step 3: find the perimeter
The side length is equivalent for all 3 sides so
P = 3(AC)
[tex]=3(8\sqrt{6})[/tex]
[tex]=24\sqrt{6}[/tex]
Step 4: find the area
[tex]A=\dfrac{1}{2}b \cdot h[/tex]
[tex]=\dfrac{1}{2}(8\sqrt6)(12\sqrt2)[/tex]
[tex]=(48\sqrt{12})[/tex]
[tex]=(96\sqrt3)[/tex]
