Select two positive lengths to represent two side lengths of a triangle.
Use a ruler to draw several different possibilities for triangle so that the length of and the length of .
What connection do you see between angle and the length of segment in the triangles that you drew?
What else do you notice about the length of segment ?
15.2
Activity
Is this triangle a right triangle? Explain or show your reasoning.
Triangle is constructed so that , , and . What can you say about the angles in triangle ? Explain or show your reasoning.
Triangle is constructed so that , , and . What can you say about the angles in triangle ? Explain or show your reasoning.
A triangle has sides of length and so that .
Make a conjecture about a triangle for which .
Make another conjecture about a triangle for which .
15.3
Activity
Triangle is constructed so that and . Is there a maximum length of ? Explain your reasoning.
Is it possible for a triangle to have sides of length 1, 16, and 20? Explain or show your reasoning.
A triangle has sides of length and . Write an inequality that shows the maximum length for .
Student Lesson Summary
The Pythagorean Theorem describes a relationship among the side lengths of a right triangle. In particular, if the sides are labeled and with , as the longest side, then
The converse of this theorem is also true. In fact, if we know the lengths of all three sides, we can classify the triangle as either acute, right, or obtuse.
If , then the triangle has a right angle.
If , then the triangle has an obtuse angle.
If , then the triangle has only acute angles.
A key reason this is true is due to the relationship between an angle and the side opposite the angle in a triangle. If the two segments making up the angle are constant, then when the angle is made greater, the opposite side must also be greater.
Another relationship among the sides of a triangle is due to limitations from the ways the ends of the segments can be connected. The length of any side of a triangle must be less than the sum of the other two side lengths in the triangle and must be greater than the positive difference between the other two side lengths. For a triangle with side lengths and , this can be written as
If the third side is too long, then putting the other two sides end to end will not be enough to reach both ends of the third side. If the third side is too short, then putting it at the end of one of the other sides will not be enough to reach the end of the other side.
