a black bordering sapphire street is a right triangle. you start walking around the block, taking 125 places on sapphire street and 102 paces on diamond street.
a.)At what angle do diamond and sapphire street interest ?
b.)How Many Paces must you take on Gold Street to complete the trip ?
To solve this problem, we need to use the concept of right triangles and trigonometry. Let's start with the given information:
- A black bordering sapphire street is a right triangle.
- You start walking around the block, taking 125 paces on sapphire street and 102 paces on diamond street.
a.) To find the angle at which diamond and sapphire street intersect, we can apply the inverse tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this case, diamond street is the opposite side, and sapphire street is the adjacent side.
First, let's define the variables:
Opposite side (diamond street): 102 paces.
Adjacent side (sapphire street): 125 paces.
Let θ be the angle.
Using the formula for the tangent function, we have:
tan(θ) = Opposite/Adjacent
tan(θ) = 102/125
Now, to find θ, we take the inverse tangent (also known as arctan) of both sides:
θ = arctan(tan(θ))
θ = arctan(102/125)
Using a scientific calculator or an online trigonometric calculator, you can find the approximate value of θ. Note that the result will be in radians. If you need the angle in degrees, you can convert it by multiplying the radian measure by (180/π).
b.) To determine the number of paces you would need to take on Gold Street to complete the trip, we can use the Pythagorean theorem. In a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
Let's assume the hypotenuse is Gold Street. We know the lengths of the other two sides: Sapphire Street = 125 paces and Diamond Street = 102 paces.
Using the Pythagorean theorem:
Hypotenuse² = Sapphire Street² + Diamond Street²
Gold Street² = 125² + 102²
Now we can take the square root of both sides to find the length of Gold Street (the hypotenuse):
Gold Street = √(125² + 102²)
Calculating this expression will give you the number of paces you need to take on Gold Street to complete the trip.