A piece of wire is bent into the shape of an isosceles triangle. The base angles are each 48 degree and the perpendicular height to the base is 6 cm. Calculate the length of the wire.

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Assistance and way to solve it

To find the length of the wire, we can start by building a triangle using the given information.

Step 1: Draw the triangle
First, draw an isosceles triangle with two equal sides and two equal base angles. Label the base angles as 48 degrees and denote the length of the equal sides as "x". The perpendicular height to the base is given as 6 cm.

/\
/ \
/____\

|______|
6 cm
|______|

Step 2: Find the length of the base
Since the base angles are equal, we know that each base angle is 48 degrees. Therefore, the sum of the base angles is 48 + 48 = 96 degrees.

Subtracting this sum from the total angle of a triangle (180 degrees) gives us the third angle: 180 - 96 = 84 degrees.

Since we have an isosceles triangle, the third angle is also 84 degrees, meaning the angle opposite the base is 84 degrees.

Step 3: Apply the Sine Rule
Using the Sine Rule, we can relate the lengths of the sides and the opposite angles in a triangle:

a/sinA = b/sinB = c/sinC

Where a, b, and c are the lengths of the sides, and A, B, and C are the angles opposite to them. In our case, a = 6 cm, A = 84 degrees, and B = 48 degrees.

So, we have:
6/sin84 = b/sin48

Rearranging the equation to solve for b:
b = (6 * sin48) / sin84

Using a scientific calculator, evaluate the right side of the equation.

Step 4: Calculate the length of the wire
The length of the wire is equal to the sum of the three sides of the triangle. In our case, the equal sides have a length of x, and the base has a length of b (which we just calculated). So:

Length of the wire = 2x + b

Substitute the value of b that we found into the equation and solve for the length of the wire.

If the base is b, then

(b/2)/6 = tan 48°
Then each of the two equal sides s is
6/s = sin 48°

Now the perimeter is just b+2s