SOLUTION: Two streets meet at an angle of 52 degrees. If a triangular lot has frontages of 60 m and 65 m on the two streets, what is the perimeter of the lot?

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Question 279149: Two streets meet at an angle of 52 degrees. If a triangular lot has frontages of 60 m and 65 m on the two streets, what is the perimeter of the lot?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
I don't know if there's an easier way to do this, but this is the way I did it.

Let the intersection of the two streets be point A.

Let the other end of the top street be point B so that the street is represented by the line segment AB.

Let the other end of the bottom street be point C so that the street is represented by the line segment AC.

You have a triangle called ABC where:

angle A is 52 degrees.
Line Segment AB is 60 meters.
Line Segment AC is 65 meters.

We drop a perpendicular from point B to intersect with line AC at point D.

We now have 2 triangles.

They are ABD and BDC.

Since we know angle A and we know AB = 60 meters, we can find the length of the line segment BD using the formula:

Sin(52) = BD / AB which becomes:

Sin(52) = BD / 60 because BD = 60 meters.

We solve for BD to get:

BD = 60 * Sin(52) which becomes:

BD = 47.28064522

We can also solve for AD using the formula:

Cos(52) = AD / AB which becomes:

Cos(52) = AD / 60 because AB = 60 meters.

We solve for AD to get:

AD = 60 * Cos(52) which becomes:

AD = 36.93968852

We now have AB and AD and BD and AC.

AB and AC were given.
AD and BD were solved for.

We know that AC = 65 meters and we know that AC = AD + DC and we know that AD = 36.93968852 so we can solve for BC using the formula:

DC = AC - AD which becomes:

DC = 65 - 36.93968852 which becomes:

DC = 28.06031148

Since we know BD and we know DC, we can now solve for angle C.

We use the formula:

Tan(C) = BD / DC which becomes:

Tan(C) = 47.28064522 / 28.06031148 which becomes:

Tan(C) = 1.684965088

We take the arctan(1.684965088) to find the angle.

Angle C = 59.31154245 degrees.

Now that we know angle C, we can use either BD or DC to find BC.

Either one will get the same answer.

Sin(C) = BD / BC

We solve for BC to get:

BC = BD / Sin(C) which becomes:

BC = 47.28064522 / Sin(59.341154245) which becomes:

BC = 54.98036461 meters.

We could also have used:

Cos(C) = DC / BC.

We solve for BC to get:

BC = DC / Cos(C) which becomes:

BC = 28.06031148 / Cos(59.31154245) which becomes:

BC = 54.98036461 which is the same answer we got before as it should be.

Your answer is:

The perimeter of the lot is:

AB + AC + BC which becomes:

60 + 65 + 54.98036461 = 179.9803646 meters.

A picture of what I just did is shown below:

The + after the number indicates a fractional part that is not shown.

the part that's missing is AD = 36+.