the city council is planning a new city park on a triangular plot of land formed by three intersecting streets the designer placed the triangle on a grid to determine the side lengths and angle measures needed for the purchase and placement of a fence surrounding the park each unit on the grid is one inch which represents 20 feet of actual length. The measure of angle CAB in the designer's diagram is (5x+1) degrees and the measure of angle ACB is (3x+12.5) degrees. What is the actual side lengths of the triangular plot of land?
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hey everyone what's the answer
To find the actual side lengths of the triangular plot of land, we need to determine the values of x in the given angles CAB and ACB.
Let's start by setting up an equation using the fact that the sum of angles in a triangle is equal to 180 degrees.
Angle CAB + Angle ACB + Angle BAC = 180
Substituting the given expressions for the angles, we have:
(5x+1) + (3x+12.5) + Angle BAC = 180
Combining like terms:
8x + 13.5 + Angle BAC = 180
Now, let's find the value of Angle BAC. Since the triangle is formed by three intersecting streets, we know that the sum of the three angles surrounding each point of intersection is 360 degrees.
So, Angle BAC = 360 - (Angle CAB + Angle ACB)
Substituting the given expressions for the angles:
Angle BAC = 360 - ((5x+1) + (3x+12.5))
Simplifying:
Angle BAC = 360 - (8x + 13.5)
Angle BAC = 360 - 8x - 13.5
Angle BAC = 346.5 - 8x
Now, substitute the value of Angle BAC back into the equation:
8x + 13.5 + (346.5 - 8x) = 180
Simplifying:
8x + 13.5 + 346.5 - 8x = 180
360 = 180
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