Question

two ropes are tied to the same place on a tree growing perpendicular to the ground. each rope is anchored to the ground the same distance from the base of the tree. which statement is true because of the perpendicular bisector theorem? a. for each anchor, the length of the rope between the anchor and where the rope is tied to the tree is the same as the distance between the anchor and the base of the tree. b. the ropes form right angles where they are anchored to the ground. c. the length of rope between where it is tied to the tree to where it is anchored to the ground is the same for both ropes. d. the height at which the ropes are tied to the tree is the same distance as the length between the two anchors

Explanation:

Step1: Recall the perpendicular bisector theorem

The perpendicular bisector theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. In this case, the tree (where the ropes are tied) is on the perpendicular bisector of the segment between the two anchors.

Step2: Analyze each option

• Option A: For each anchor, the length of the rope between the anchor and where the rope is tied to the tree is the same as the distance between the anchor and the base of the tree. This is incorrect. The length of the rope (hypotenuse in the right - triangle formed by the tree height, distance from anchor to base of tree) is longer than the distance from the anchor to the base of the tree (one of the legs of the right - triangle).
• Option B: The ropes form right angles where they are anchored to the ground. There is no information in the perpendicular bisector theorem that implies the ropes form right angles at the anchor.
• Option C: The length of rope between where it is tied to the tree to where it is anchored to the ground is the same for both ropes. Since the tree is on the perpendicular bisector of the segment between the two anchors, and using the Pythagorean theorem \(l=\sqrt{h^{2}+d^{2}}\) (where \(h\) is the height of the tie - point on the tree and \(d\) is the distance from the anchor to the base of the tree, and \(d\) is the same for both anchors as the tree is on the perpendicular bisector), the length of the ropes (the hypotenuse) is the same.
• Option D: The height at which the ropes are tied to the tree is the same distance as the length between the two anchors. There is no geometric relationship (from the perpendicular bisector theorem) that equates the height of the tie - point on the tree to the distance between the two anchors.

Answer:

C. The length of rope between where it is tied to the tree to where it is anchored to the ground is the same for both ropes.