We need a piece of wood that is approximately 8.1 inches long to replace AD. So, the answer is option A.
Describe Triangle?
In geometry, a triangle is a three-sided polygon consisting of three line segments that connect three non-collinear points. These three points are called the vertices of the triangle, and the line segments connecting them are called sides. The sum of the angles of a triangle is always 180 degrees.
Triangles can be classified based on their sides and angles. A triangle with all three sides of equal length is called an equilateral triangle, while a triangle with two sides of equal length is called an isosceles triangle. A triangle with no sides of equal length is called a scalene triangle.
Triangles can also be classified based on their angles. A triangle with all three angles less than 90 degrees is called an acute triangle, while a triangle with one angle greater than 90 degrees is called an obtuse triangle. Finally, a triangle with one angle equal to 90 degrees is called a right triangle.
First, we can use trigonometry to find the length of side AC. Since angle BCD is 30 degrees and BC = 14, we have:
sin(30) = AC/14
AC = 14*sin(30)
AC = 7 inches
Next, we can use the fact that triangle ACD is a 15-75-90 triangle to find the length of AD. Since angle CAD is 75 degrees, we have:
tan(75) = AD/AC
AD = ACtan(75)
AD = 7tan(75)
Using a calculator, we get:
AD ≈ 8.09 inches
Therefore, we need a piece of wood that is approximately 8.1 inches long to replace AD. So, the answer is option A.
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