A right triangle with an angle at the ground calls for the tangent ratio: the side across from the angle divided by the ground-level side touching it. If that ground side is missing, check whether the given lengths belong to a familiar right-triangle triple. Name the sides from the marked angle before forming the ratio. The rope can touch the angle without being the side tangent uses.
Hints
- Hint 1
The hypotenuse is the side across from the right angle. Identify it first; although the rope touches , touching the angle doesn’t make it the adjacent leg.
- Hint 2
A Pythagorean triple is a set of whole-number side lengths that makes a right triangle. The given lengths fit a familiar triple. What is the missing ground length?
- Hint 3
Tangent means opposite leg divided by adjacent leg. From the angle where the rope touches the ground, which leg is across from the angle, and which runs along the ground?
Step-by-step
Use the right-triangle sides and tangent
Step 1Identify the hypotenuse
No Desmos needed. A familiar right-triangle triple gives the missing side. The right angle is where the vertical pole meets the ground, so the rope is the hypotenuse, the side across from that angle. Its length is meters.
- Step 2
Find the ground length
The pole is meters, and , , and form a Pythagorean triple: . So the ground leg is meters.
- Step 3
Name the legs from the marked angle
From on the ground, the -meter pole is the opposite leg, across from the angle. The -meter ground segment is the adjacent leg, which touches the angle. Don’t use the rope as adjacent: it’s the hypotenuse.
- Step 4
Form the tangent ratio
Tangent is opposite divided by adjacent, so . The angle between the rope and the ground has tangent . Choice A.