7.2.4: Drawing Triangles (Part 1)

Lesson

Let's see how many different triangles we can draw with certain measurements.

Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Triangles

Which one doesn't belong?

Figure \(\PageIndex{1}\): Four triangles, A, B, C, D. Triangle A has 3 60 degree angles and 2 sides with length 3. Triangle B has angles labeled 130 degrees, 20 degrees, and the side between them length 7. Triangle C has 2 sides length 9, angle 55 degrees. Triangle D has one angle marked 90 degrees.

Exercise \(\PageIndex{2}\): Does Your Triangle Match Theirs?

Three students have each drawn a triangle. For each description:

1. Drag the vertices to create a triangle with the given measurements.
2. Make note of the different side lengths and angle measures in your triangle.
3. Decide whether the triangle you made must be an identical copy of the triangle that the student drew. Explain your reasoning.

Jada’s triangle has one angle measuring 75°.

Andre’s triangle has one angle measuring 75° and one angle measuring 45°.

Lin’s triangle has one angle measuring 75°, one angle measuring 45°, and one side measuring 5 cm.

Exercise \(\PageIndex{3}\): How Many Can You Draw?

1. Draw as many different triangles as you can with each of these sets of measurements:
   1. Two angles measure \(60^{\circ}\), and one side measures 4 cm.
   2. Two angles measure \(90^{\circ}\), and one side measures 4 cm.
   3. One angle measures \(60^{\circ}\), one angle measures \(90^{\circ}\), and one side measures 4 cm.
2. Which sets of measurements determine one unique triangle? Explain or show your reasoning.

Are you ready for more?

Figure \(\PageIndex{2}\)

In the diagram, 9 toothpicks are used to make three equilateral triangles. Figure out a way to move only 3 of the toothpicks so that the diagram has exactly 5 equilateral triangles.

Summary

Sometimes, we are given two different angle measures and a side length, and it is impossible to draw a triangle. For example, there is no triangle with side length 2 and angle measures \(120^{\circ}\) and \(100^{\circ}\):

Figure \(\PageIndex{3}\): In the figure a horizontal line segment is drawn and labeled 2. On the left end of the line segment, a dashed line is drawn upward and to the left. The angle formed between the dashed line and the horizontal line is labeled 120 degrees. On the right end of the horizontal line, a dashed line is drawn upward and to the right. The angle formed between the dashed line and horizontal line is labeled 100 degrees.

Sometimes, we are given two different angle measures and a side length between them, and we can draw a unique triangle. For example, if we draw a triangle with a side length of 4 between angles \(90^{\circ}\) and \(60^{\circ}\), there is only one way they can meet up and complete to a triangle:

Figure \(\PageIndex{4}\)

Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and same angle measures.

Practice

Exercise \(\PageIndex{4}\)

Use a protractor to try to draw each triangle. Which of these three triangles is impossible to draw?

1. A triangle where one angle measures \(20^{\circ}\) and another angle measures \(45^{\circ}\)
2. A triangle where one angle measures \(120^{\circ}\) and another angle measures \(50^{\circ}\)
3. A triangle where one angle measures \(90^{\circ}\) and another angle measures \(100^{\circ}\)

Exercise \(\PageIndex{5}\)

A triangle has an angle measuring \(90^{\circ}\), an angle measuring \(20^{\circ}\), and a side that is 6 units long. The 6-unit side is in between the \(90^{\circ}\) and \(20^{\circ}\) angles.

1. Sketch this triangle and label your sketch with the given measures.
2. How many unique triangles can you draw like this?

Exercise \(\PageIndex{6}\)

1. Find a value for \(x\) that makes \(-x\) less than \(2x\).
2. Find a value for \(x\) that makes \(-x\) greater than \(2x\).

(From Unit 5.4.1)

Exercise \(\PageIndex{7}\)

One of the particles in atoms is called an electron. It has a charge of -1. Another particle in atoms is a proton. It has charge of +1.

The overall charge of an atom is the sum of the charges of the electrons and the protons. Here is a list of common elements.

Table \(\PageIndex{1}\)

	charge from electrons	charge from protons	overall charge
carbon	\(-6\)	\(+6\)	\(0\)
aluminum	\(-10\)	\(+13\)
phosphide	\(-18\)	\(+15\)
iodide	\(-54\)	\(+53\)
tin	\(-50\)	\(+50\)

Find the overall charge for the rest of the atoms on the list.

(From Unit 5.2.2)

Exercise \(\PageIndex{8}\)

A factory produces 3 bottles of sparkling water for every 7 bottles of plain water. If those are the only two products they produce, what percentage of their production is sparkling water? What percentage is plain?

(From Unit 4.1.3)