A road’s rise and horizontal run form the two legs of a right triangle. For the angle the road makes with the horizontal, tangent is rise over run: the side across from the angle divided by the leg along the horizontal. Read the lengths relative to that angle, then reduce the fraction. Don’t use the slanted road as the denominator.
Hints
- Hint 1
Stand at the angle where the road meets the horizontal. The opposite leg is across from that angle, while the adjacent leg touches it along the horizontal. Which given length is each leg?
- Hint 2
The tangent of an angle is opposite divided by adjacent, so use rise over horizontal distance. After writing the fraction, check whether its numerator and denominator share a factor.
Step-by-step
Use rise over run
Step 1Identify the two legs from the angle
No Desmos needed. The two given lengths make a fraction you can reduce by hand. From at the bottom of the road, the -meter rise is opposite, or across from the angle. The -meter horizontal distance is adjacent, the leg touching the angle. The slanted road isn’t either of those legs.
- Step 2
Write the tangent ratio
The tangent of an angle is opposite divided by adjacent, so:
For an angle measured from the horizontal, tangent compares the rise with the horizontal run.
- Step 3
Reduce to simplest form
The question asks for simplest form. Since and are both divisible by , divide the top and bottom by :
So the road’s rise-to-run ratio is . Choice A.