A block 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. At what angle do Diamond and Sapphire Streets intersect?
b. How many paces must you take on Gold Street to complete the trip?
I cannot make sense out of your description.
Where doe Gold Street come from?
To solve this problem, we can use trigonometry and the Pythagorean theorem. Let's start by solving part a.
a. At what angle do Diamond and Sapphire Streets intersect?
Step 1: Draw a diagram of the right triangle and label the sides.
```
|\
| \
125 | \
| \ c
| \
| \
| \
|-------\
102 a
```
We have a right triangle with sides of length 125, 102, and c. The angle between Sapphire Street and Diamond Street is represented by the angle a.
Step 2: Use the inverse tangent function (tan^(-1)) to calculate the angle.
tan(a) = opposite/adjacent
tan(a) = 125/102
a = tan^(-1)(125/102)
Using a scientific calculator or math software, we can find:
a ≈ 50.73 degrees
So, Diamond and Sapphire Streets intersect at an angle of approximately 50.73 degrees.
Now, let's move on to part b.
b. How many paces must you take on Gold Street to complete the trip?
Step 1: Calculate the hypotenuse of the right triangle using the Pythagorean theorem.
c^2 = 125^2 + 102^2
c^2 = 15625 + 10404
c^2 = 26029
c ≈ √26029
Using a calculator or math software, we find:
c ≈ 161.19 paces (rounded to two decimal places)
Step 2: Find the length of Gold Street.
Gold Street completes the triangle, so its length is equal to the hypotenuse.
Therefore, you must take approximately 161.19 paces on Gold Street to complete the trip.
Keep in mind that these calculations assume the streets form a perfect right triangle, and the given measurements are accurate.
To solve these questions, we can use the concept of trigonometry and the Pythagorean theorem. Let's break it down step by step:
a. To find the angle at which Diamond and Sapphire Streets intersect, we can use the tangent function. The tangent of an angle is defined as the ratio of the length of the opposite side of the right triangle (here it is Diamond Street) to the length of the adjacent side (here it is Sapphire Street).
First, let's find the length of the two sides of the right triangle using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
We have:
a^2 + b^2 = c^2
Given that Sapphire Street is 125 paces and Diamond Street is 102 paces, we can substitute these values into the equation:
125^2 + 102^2 = c^2
Solving this, we get:
15625 + 10404 = c^2
26029 = c^2
c ≈ √26029
c ≈ 161.28 paces
So the length of the hypotenuse, which represents the distance between the starting point and the end point of the path, is approximately 161.28 paces.
Now let's find the tangent of the angle:
tan(θ) = (opposite side) / (adjacent side)
tan(θ) = 102 / 125
To find the angle itself, we can calculate the inverse tangent (arctan) of the ratio:
θ = arctan(102 / 125)
Using a calculator, we get:
θ ≈ 40.64 degrees
Therefore, Diamond and Sapphire Streets intersect at an angle of approximately 40.64 degrees.
b. To find how many paces you must take on Gold Street to complete the trip, we can use the Pythagorean theorem again.
We know the length of Diamond Street is 102 paces and the length of Sapphire Street is 125 paces. The length of the hypotenuse, as we calculated earlier, is approximately 161.28 paces.
Since we are walking around the block, the length of Gold Street represents the remaining portion of the path.
To find the length of Gold Street, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
Let's assume the length of Gold Street is 'x' paces:
x^2 + 125^2 = 161.28^2
Simplifying this equation, we get:
x^2 + 15625 = 26029
x^2 = 26029 - 15625
x^2 = 10404
x ≈ √10404
x ≈ 102 paces
Therefore, you would need to take approximately 102 paces on Gold Street to complete the trip.