Unit 8, Lesson 8
The Converse
Accelerated 7
8.1
Warm-up
Consider the tips of the hands of an analog clock that has an hour hand that is 3 centimeters long and a minute hand that is 4 centimeters long.
The image of a circle that represent an analog clock. On the circle are 12 evenly spaced tick marks. There are two hands on the clock. One hand is labeled 3, begins in the center of the circle and extends upward and to the right, and points to the third tick mark from the top. The other hand is labeled 4, begins in the center of the circle and extends upward and to the left. It points to the eleventh tick mark from the top.
Over the course of a day:
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What is the farthest distance apart the two tips get?
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What is the closest distance the two tips get?
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Are the two tips ever exactly five centimeters apart? Explain your reasoning.
8.2
Activity
Here are three triangles with two side lengths measuring 3 and 4 units, and the third side of unknown length.
A figure of three triangles each with 2 given side lengths and one unknown side length. The first triangle has a horizontal side of 4, a side length slanted upward and to the left of 3, and the third side length labeled x. The middle triangle has a horizontal side length of 4, a second side length slanted upward and to the right of 3, and the third side length labeled y. The third triangle is a right triangle with a horizontal side length of 4, a vertical side length of 3, and the third side is labeled z.
Order the following six values from smallest to largest. Put an equal sign between any you know to be equal. Be prepared to explain your reasoning.
8.3
Activity
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Given the information provided for the right triangles shown here, find the unknown leg lengths to the nearest tenth.
Two right triangles are indicated. The triangle on the left has two leg with lengths of 2 and a. The hypotenuse has a length of 7. The triangle on the right has two legs with length x and a hypotenuse of length 6. -
The triangle shown here is not a right triangle. What are two different ways you change one of the values so it would be a right triangle? Sketch these new right triangles, and clearly label the right angle.
Student Lesson Summary
How can we tell whether a triangle is a right triangle or not? For example, in this triangle it isn’t clear just by looking, and it may be that the sides aren’t drawn to scale.
If we have a triangle with side lengths , , and , with being the longest of the three, then the converse of the Pythagorean Theorem tells us that any time we have , we must have a right triangle. Since , any triangle with side lengths 8, 15, and 17 must be a right triangle.
What about the jib sail on this boat? It is a triangle, but is it a right triangle?
The measurements of the jib sail are shown here. The sum of the squares of the two shorter sides is 106.965 square meters, and the square of the longest side is 104.8576 square meters. So by the converse of the Pythagorean Theorem, it is not a right triangle, but it is close to one.
Together, the Pythagorean Theorem and its converse provide a way to check if a triangle is a right triangle just using its side lengths. If , it is a right triangle. If , it is not a right triangle.
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