For the triangle XYZ, the answers are:1)XY=[tex]8\sqrt{3}[/tex]; 2)Exact Area of triangle XYZ= [tex]32\sqrt{3} ft[/tex] and 3) Area of triangle XYZ = 55.4 ft.
RIGHT TRIANGLE
A triangle is classified as a right triangle when it presents one of your angles equal to 90º. The greatest side of a right triangle is called hypotenuse. And, the other two sides are called cathetus or legs.
The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.
The Pythagorean Theorem says: [tex](hypotenuse)^2=(leg_1)^2+(leg_2)^2[/tex] . And the main trigonometric ratios are:
[tex]sin(\alpha )=\frac{opposite \;leg}{hypotenuse} \\ \\ cos(\alpha )=\frac{adjacent\;leg}{hypotenuse}\\ \\ tan(\alpha )=\frac{opposite \;leg}{adjacent\;leg}[/tex]
- Finding the segment XY.
[tex]cos (30 )=\frac{adjacent\;leg}{hypotenuse} \\ \\ \frac{\sqrt{3} }{2} =\frac{XY}{16} \\ \\ 2XY=16\sqrt{3} \\ \\ XY=8\sqrt{3}\;ft[/tex]
2.Finding the exact area of triangle XYZ
The area of a right triangle is equal to [tex]\frac{1}{2} * base *height[/tex]. In this exercise the base is equal to [tex]8\sqrt{3}[/tex]. For finding the height, you should apply the trigonometric ratio for sin.
[tex]sin (30 )=\frac{opposite\;leg}{hypotenuse} \\ \\ \frac{{1} }{2} =\frac{XZ}{16} \\ \\ 2XZ=16\\ \\ XZ=8\;ft[/tex]
Therefore, the area will be:
[tex]\frac{1}{2} * base *height\\ \\ \frac{1}{2} * 8\sqrt{3} *8=32\sqrt{3} ft[/tex]
3.Finding the area of triangle XYZ to the nearest tenth of a square ft
Knowing that [tex]\sqrt{3}= 1.73205\dots[/tex], the solution will be:
[tex]\frac{1}{2} * base *height\\ \\ \frac{1}{2} * 8\sqrt{3} *8=32\sqrt{3} =32*1.73205=55.4 ft[/tex]
Learn more about trigonometric ratios here:
brainly.com/question/11967894
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