Answer:
The measure of side c is 16 cm.
Step-by-step explanation:
Given information: Area of triangle ABC is 24 square centimeters. [tex]\angle B=30^{\circ}[/tex] and [tex]a=6cm[/tex].
Let the height of triangle ABC is
Draw a perpendicular on BC from A.
[tex]\sin \theta=\frac{perpendicular}{hypotenuse}[/tex]
[tex]\sin B=\frac{AD}{AB}[/tex]
[tex]\sin 30=\frac{h}{c}[/tex]
[tex]\frac{1}{2}=\frac{h}{c}[/tex]
[tex]\frac{1}{2}\times c=h[/tex]
Therefore the height of triangle ABC is [tex]\frac{1}{2}\times c[/tex].
The area of triangle ABC is
[tex]A=\frac{1}{2}\times base\times height[/tex]
[tex]A=\frac{1}{2}\times BC\times AD[/tex]
[tex]A=\frac{1}{2}\times a\times h[/tex]
[tex]A=\frac{1}{2}\times 6\times (\frac{1}{2}\times c)[/tex]
[tex]A=3\times (\frac{c}{2})[/tex]
[tex]A=\frac{3c}{2}[/tex]
The area of triangle ABC is 24 square centimeters.
[tex]24=\frac{3c}{2}[/tex]
Multiply 2 both sides.
[tex]48=3c[/tex]
Divide both sides by 3.
[tex]c=16[/tex]
Therefore the measure of side c is 16 cm.
