Question

two wheelchair ramps, each 10 feet long, lead to the two ends of the entrance porch of mr. bell’s restaurant. the two ends of the porch are at the same height from the ground, and the start of each ramp is the same distance from the base of the porch. the angle of the first ramp to the ground is 24°. which statement must be true about the angle of the second ramp to the ground? a. nothing is known about the angle of the second ramp. b. it must have an angle of exactly 24°. c. it could have any angle greater than or equal to 24°. d. it could have any angle less than or equal to 24°.

Explanation:

Step1: Analyze the given information

We have two wheelchair ramps, each 10 feet long. The two ends of the porch are at the same height from the ground, and the start of each ramp is the same distance from the base of the porch. Let's consider the right - triangle formed by each ramp, the ground, and the vertical height from the ground to the porch.
For a right - triangle with hypotenuse (length of the ramp) \(c = 10\) feet, the height \(h\) from the ground to the porch and the horizontal distance \(d\) from the start of the ramp to the base of the porch can be related to the angle of the ramp with the ground (\(\theta\)) using the trigonometric ratios. The height \(h=c\sin\theta\) and the horizontal distance \(d = c\cos\theta\).
Since the height \(h\) (because the two ends of the porch are at the same height) and the horizontal distance \(d\) (because the start of each ramp is the same distance from the base of the porch) are the same for both ramps, and the hypotenuse \(c = 10\) feet is also the same for both ramps.

Step2: Use the congruence of right - triangles

In two right - triangles, if the hypotenuse and one leg (either the height or the horizontal distance) are equal, then the triangles are congruent by the Hypotenuse - Leg (HL) congruence criterion.
For the first ramp, let the angle with the ground be \(\theta_1=24^{\circ}\), hypotenuse \(c_1 = 10\) feet, height \(h_1 = 10\sin\theta_1\) and horizontal distance \(d_1=10\cos\theta_1\).
For the second ramp, hypotenuse \(c_2 = 10\) feet, height \(h_2=h_1\) (same height of porch) and horizontal distance \(d_2 = d_1\) (same distance from start of ramp to base of porch). So, the two right - triangles formed by the ramps, ground and porch are congruent.
Since the triangles are congruent, the angles of the ramps with the ground must be equal. So the angle of the second ramp with the ground must also be \(24^{\circ}\).

Answer:

B. It must have an angle of exactly \(24^{\circ}\)