Activity 1.1. Properties of Triangles.
- Materials: You will need paper and pencil, scissors, a ruler and compass, and two plastic straws.
Prepare: Cut the straw into pieces of lengths [latex]2[/latex] inches, [latex]3[/latex] inches, and [latex]6[/latex] inches. Use the ruler to draw a large triangle, and cut it out.
- What do we know about the sides of a triangle?
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- Can you make a triangle with sides of length [latex]2[/latex] inches, [latex]3[/latex] inches, and [latex]6[/latex] inches?
- Use the pieces of length [latex]2[/latex] inches and [latex]3[/latex] inches to form two sides of a triangle. What happens to the length of the third side as you increase the angle between the first two sides?
- What is the longest that the third side could be? What is the smallest?
- Two sides of a triangle are [latex]6[/latex] centimeters and [latex]8[/latex] centimeters long. What are the possible lengths of the third side?
- Two sides of a triangle are [latex]p[/latex] units and [latex]q[/latex] units long. What are the possible lengths of the third side?
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- What do we know about the angles of a triangle?
- Use a protractor to measure the three angles of the paper triangle in degrees. Now add up the three angles. What is their sum?
- Tear off the three corners of the triangle. Place them side-by-side with their vertices (tips) at the same point. What do you find?
- How are your answers to parts ([latex]1[/latex]) and ([latex]2[/latex]) related?
- How are the sides and angles of a triangle related?
- A standard way to label a triangle is to call the angles [latex]A,~B,[/latex] and [latex]C\text{.}[/latex] The side opposite angle [latex]A[/latex] is called [latex]a\text{,}[/latex] the side opposite angle [latex]B[/latex] is called [latex]b\text{,}[/latex] and the side opposite angle [latex]C[/latex] is called [latex]c\text{.}[/latex] Sketch a triangle and label it with standard notation.
- Using a ruler, carefully draw a triangle and label it with standard notation so that [latex]a \gt b \gt c\text{.}[/latex] Now use a protractor to measure the angles and list them in order from largest to smallest. What do you observe?
- Using a ruler, carefully draw another triangle and label it with standard notation so that [latex]A \gt B \gt C\text{.}[/latex] Now use a ruler to measure the sides and list them in order from largest to smallest. What do you observe?
- What do we know about right triangles?
- The side opposite the [latex]90°[/latex] angle in a right triangle is called the hypotenuse. Why is the hypotenuse always the longest side of a right triangle?
- The Pythagorean theorem states that:
If: [latex]a\text{,}[/latex] [latex]b\text{,}[/latex] and [latex]c[/latex] are the sides of a right triangle, and [latex]c[/latex] is the hypotenuse, THEN: [latex]\qquad[/latex][latex]a^{2}+b^{2}=c^{2}[/latex] The “if” part of the theorem is called the hypothesis, and the “then” part is called the conclusion. The converse of a theorem is the new statement you obtain when you interchange the hypothesis and the conclusion. Write the converse of the Pythagorean theorem. - The converse of the Pythagorean theorem is also true. Use the converse to decide whether each of the following triangles is a right triangle. Support your conclusions with calculations.
- [latex]\displaystyle {a = 9,~ b = 16,~ c = 25}[/latex]
- [latex]\displaystyle {a = 12,~ b = 16,~ c = 20}[/latex]
- [latex]\displaystyle {a = \sqrt{8},~ b = \sqrt{5},~ c = \sqrt{13}}[/latex]
- [latex]\displaystyle {a = \frac{\sqrt{3}}{2},~ b = \frac{1}{2},~ c = 1}[/latex]