A black bordering Sapphire St. is a right triangle. You start walking around the block, taking 125 paces on sapphire st. and 102 paces on Diamond st.

a.) At what angle do Diamond and Sapphire Street Intersect ?

To find the angle at which Diamond and Sapphire Street intersect, we can use properties of right triangles and trigonometry.

Step 1: Plotting the streets
First, let's visualize the streets on a coordinate grid. Assume Sapphire Street runs horizontally along the x-axis, and Diamond Street runs vertically along the y-axis. Mark the starting point at the origin (0,0).

Step 2: Plotting the triangle
Next, plot the triangle with Sapphire Street as the base and Diamond Street as the height. The point where the two streets intersect will be one of the vertices of the right triangle. Mark this point on the grid.

Step 3: Calculating the angle
To find the angle, we need to determine the opposite and adjacent sides of the triangle. In this case, the adjacent side is the length of Sapphire Street, which is 125 paces, and the opposite side is the length of Diamond Street, which is 102 paces.

Step 4: Applying trigonometry
Now, we can use the tangent function (tan) to find the angle. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side.

tan(angle) = opposite/adjacent
tan(angle) = 102/125

To find the angle, we can take the arctangent (inverse tangent) of the calculated value:

angle = arctan(102/125)

Using a calculator, we find that arctan(102/125) is approximately 42.1 degrees.

Therefore, Diamond and Sapphire Streets intersect at an angle of approximately 42.1 degrees.