Answer:
a = 14
b= 6√2
Step-by-step explanation:
Here we can divide this shape into a rectangle and a triangle as shown in the attachment
- Since the rectangle has a height of 6, our line we drew dividing the two shapes must mean that the height of our triangle is also 6
- We will need to figure out the missing sides of the triangle first before solving for a
b:
- We cannot yet use the Pythagorean Theorem because we only have one leg of the triangle.
- However, we can use trig identities to solve for one of the missing legs since we were given an angle.
- We will want to use and identity that involves the "opposite" leg and the "hypotenuse"; this identity is sin: [tex]sin(\alpha ) =\frac{opposite}{hypotenuse}[/tex]
- Solving for the hypotenuse, b, our equation becomes [tex]b=\frac{opposite}{sin(\alpha )}[/tex]
- Then plugging in our known values we have [tex]b=\frac{6}{sin(45)}=\frac{6}{(\frac{\sqrt{2} }{2} )} = 6\sqrt{2}[/tex]
- b = 6√2
- Now that we have solve for two legs, we can use the Pythagorean theorem to solve for the missing leg in which I labelled as "x"
- a² + b² = c² are the variables used in the formula, however I will change them to x² + y² = z² to avoid any confusion since there are already "a" and "b" variables used in this problem
- So here solving for x, we have [tex]x=\sqrt{z^2-y^2}[/tex]
- Then plugging in our values we get [tex]x=\sqrt{(6\sqrt{2} )^2-(6)^2}=\sqrt{72-36}=\sqrt{36}=6[/tex]
a:
- Now that we have solved for all the missing legs of the triangle, we can now solve for a
- Looking at the figure, a is 8 plus the bottom leg of the triangle
- So a = 8 + x → a = 8 + 6 → a = 14
