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Outside the Box Geometry

# [b]ASS Fishing

This quiz will explore descriptions of triangles… and fishing.

As in previous quizzes, we will be looking at descriptions of triangles, each of which will do one of the following:

• define a unique triangle, if all triangles that have all of the specified properties are congruent;

• underconstrain a triangle, if at least two different (non-congurent) triangles can have all of the specified properties;

• overconstrain a triangle, if no triangle exists that has all of the specified properties.

One type of triangle that will come up again and again in this quiz is the 30-60-90 right triangle. You can think of this triangle as half of an equilateral triangle.

If we assume that the shortest side in this triangle has a side length of 1, then the hypotenuse will have a length of 2. What is the length of the other leg?

Which of these sets of edges does not describe a 30-60-90 triangle?

Hint: Remember that any 30-60-90 triangle will have either the three leg lengths $$\big(1, \sqrt{3}, 2\big)$$ or the lengths will all be scaled larger or smaller together. For example, a triangle with side lengths $$\big(2, 2\sqrt{3}, 4\big)$$ will also be a 30-60-90 triangle.

Sally wants to make a right triangle with one angle of $$30^\circ,$$ one side length of 12, and another side length of 4. Is it possible?

Let's take this knowledge about triangles and go fishing. You attach your rod to the shore at the fixed angle of $$30^\circ,$$ indicated by the red angle, and are trying to determine how far to let the line out. The yellow bob that you’re lowering will float on the surface of the water.

Case 1: If the line isn't long enough, it won't touch the water, and a triangle won't be created.

Case 2: If the line is longer, it will touch down exactly to the water beneath the tip of the rod, creating a right triangle.

Case 3: If the line is longer still, it could touch down on the water at two different places, creating two different triangles.

Case 4: If the line is longer than the rod, then there is only one angle at which the line could meet the water.

Your fishing pole is 10 ft long. If you want to keep that fixed $$30^\circ$$ angle, what is the minimum length of line, in feet, that you must let out so that your line reaches the water?

Let's examine another scenario. Your 10 ft fishing pole is still attached to the ground at a $$30^\circ$$ angle. Unfortunately, due to a mishap with your pole, your line is exactly 8 ft long, without any ability to extend or contract. If the line is drawn taut by currents, how far from the base of the rod will the bob come to rest?

In the previous problem, we were given two side lengths and one angle adjacent to one side, but not either of the other angles.

We found that this description $$\text{__________}.$$

A. uniquely defined a triangle
B. was underconstrained
C. was overconstrained

ASS is one of the most complicated triangle description types to understand. “A-S-S” represents the scenario of angle-side-side, a description of two of a triangle’s side lengths and one angle that is not the angle between the two sides.

Depending on the specified angle and side lengths, an ASS description can be uniquely defining, or overconstraining, or underconstraining. Here is an example of each:

True or False?

In all of the cases where an ASS description is uniquely defining, the triangle is a 30-60-90 triangle.

So far, studying ASS, we’ve looked at 3 cases for the lengths of the line:

1) The line is too short to touch the water.
2) The line is just long enough to touch the water.
3) The line is longer than case 2 but shorter than the length of the pole.

This question addresses case 4, in which the line is longer than the length of the pole.

Say you’re fishing from the shore of a beach with your rod planted at a $$30^\circ$$ angle relative to the ground and your line gets stuck at 15 ft long.

Does this ASS description with one angle two sides overconstrain, underconstrain, or uniquely define a triangle?

As we have seen in the previous examples, a description of angle-side-side, or ASS, can be an underconstraining description of a triangle.

Let's examine a similar scenario in which the description is ASA, or angle-side-angle.

If a triangle has one $$20^\circ$$ angle, one $$60^\circ$$ angle, and an included side length of 10, is the triangle uniquely defined?

We have seen that we need to define three parameters of a triangle to uniquely define it. However, not any three parameters will work.

SSS, ASA, SAS, and AAS will completely define a unique triangle.

ASS, or angle-side-side, can either underconstrain, overconstrain, or uniquely define a triangle. As we saw in the previous problem, there are times when one specified angle and two side lengths can create more than one triangle, exactly one triangle, or no possible triangle.

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