Given:
The figure of a right angle triangle with legs a and 7, and hypotenuse c.
The measure of angle opposite to leg 7 is 30^\circ.
To find:
All the missing measures for the triangle.
Solution:
According to the angle sum property, the sum of all interior angles of a triangle is 180 degrees. So,
[tex]30^\circ+90^\circ+\text{Missing angle}=180^\circ[/tex]
[tex]120^\circ+\text{Missing angle}=180^\circ[/tex]
[tex]\text{Missing angle}=180^\circ-120^\circ[/tex]
[tex]\text{Missing angle}=60^\circ[/tex]
In a right angle triangle ,
[tex]\sin\theta=\dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex]\sin (30^\circ)=\dfrac{7}{c}[/tex]
[tex]\dfrac{1}{2}=\dfrac{7}{c}[/tex]
[tex]c=7\times 2[/tex]
[tex]c=14[/tex]
In a right angle triangle,
[tex]\tan \theta =\dfrac{Perpendicular}{Base}[/tex]
[tex]\tan (30^\circ)=\dfrac{7}{a}[/tex]
[tex]\dfrac{1}{\sqrt{3}}=\dfrac{7}{a}[/tex]
[tex]a=7\times \sqrt{3}[/tex]
[tex]a=7\sqrt{3}[/tex]
Therefore, the measure of the missing angle is 60 degrees. Side "a" has a length of 14 units and side "c" has a length of [tex]7\sqrt{3}[/tex] units.
