Pythagorean Theorem and Converse Worksheets
When we are working with a triangle that has a right angle we can use the Pythagorean Theorem to determine the length of any of the sides, if we know the two other measures. It can be described as a2 + b2 = c2. Where c is the measure of the longest side called the hypotenuse. The other two sides are described as a and b respectively. You will use this countless times to determine the measure of missing sides, but if you look at this theorem in reverse it can be used to determine the classification of a triangle altogether. We call this the converse of the Pythagorean Theorem. We do this by comparing the sum of the squares of the shorter sides with the square of the hypotenuse. If they are equal, you have a right triangle. If the sum of the squares of the shorter are larger than square of the hypotenuse than you have an acute triangle. If the opposite is true, you have an obtuse triangle. These worksheets will help you test the use of the converse of the Pythagorean Theorem in a variety of situations.
Aligned Standard: Grade 8 Geometry - 8.G.B.6
- Cloth Triangle Step-by-Step Lesson - I really like the way this skill can be applied to real world problems like this one.
- Guided Lesson - These are all thick word problems that I would encourage students to draw before they start on.
- Guided Lesson Explanation - This really helps bring the theorem to light.
- Independent Practice - A string of problems that I would start by drawing out and visualizing for yourself.
- Matching Worksheet - These are all well written problems that you will see on a test some day soon.
- Pythagorean Theorem Worksheet Five Pack - These are the great old problems people think of as word problems. A train leaves...
- Pythagorean Theorem Worksheet Five Pack Version 2 - Half word problems and half in your face triangles.
- Answer Keys - These are for all the unlocked materials above.
These problems really test students to see if they truly understand the concept and use of Pythagorean theorem.
- Homework 1 - A triangle shaped piece of chocolate is 3 inches long and 5 inches wide. How long is the diagonal of triangle?
- Homework 2 - A garden is in the shape of a triangle and has sides with the lengths of 5 kilometers, 8 kilometers and 14 kilometers. Find out if it is a right triangle?
- Homework 3 - A triangular shaped field is 125 yards long and the length of the diagonal of the field is 150 yards. What is the width of the field?
It is best to diagram all of these problems so that you have a good handle on what is being asked of you.
- Practice 1 - Lauren leaves home to go to office. She drives 6 miles north and the she heads 8 miles east. How far is Lauren from her home?
- Practice 2 - Ellen leaves home to go to the playground. She drives 3 miles north and then heads 4 miles east. How far is Ellen from her home?
- Practice 3 - Todd is a window washer. He leaned a ladder against the side of a building. The top of the ladder reaches the window, which is 12 feet off the ground. The base of the ladder is 5 feet away from the building. How long is the ladder?
Math Skill Quizzes
Once again, diagramming is highly recommended for these.
- Quiz 1 - If the legs of an isosceles right triangle are 12 inches long, approximate the length of the hypotenuse to the nearest whole number.
- Quiz 2 - What is the length of the missing leg?
- Quiz 3 - Richard is riding a boat. He drives 12 m east and then heads to 20 m north. How far is he from his starting point?
What Is the Converse of Pythagorean Theorem?
If you look at the Pythagorean Theorem in reverse, it can be used to determine the classification of a triangle. We commonly use the Pythagorean Theorem with right triangles. It tells us that the sum of the squares of the two shorter sides is equal the square of the longest side (hypotenuse) or a2 + b2 = c2. If this balances out, you are working with a right triangle. If you look at this from a slightly different prospective, if a balance does not exist the classification of the triangle is no longer right. If the side of the equation that has the shorter sides has a larger sum than the value of the squared hypotenuse the triangle classification is acute. If that were to be flipped, you would have an obtuse triangle.
Proof: Just suppose that there is a triangle that is not right-angled. According to the Pythagoras theorem, BD2 = a2 + b2 + c2, hence the length of sides can be derived from given sides. Therefore, we now get an isosceles triangle ACD and ABD. It can be followed that we have congruent angles, CDA = CAD and BDA = DAB. But if the apparent inequalities contradict, BDA < CDA = CAD < DAB or DAB < CAD = CDA < BDA.
How Is This Skill Used Every Day?
While we have focused much of our attention on triangles in this series of lessons and worksheets it is often difficult to see how this would be used in the real world. This skill lends itself to help determine position and relative position to another point. We use navigation apps in our everyday travels. We take for granted the math behind them. When you plug in your destination and you see that measure of how far you are away from your interest and how long it will take you to get there, this math is all behind the scenes put into action. Your device and the database that it is connected to just did this math for you by finding the length of the side of a huge helping of triangles. This skill is often used by architects and anyone trying to determine a missing length. There are so many applications of this simple concept in all forms of navigation whether you are in a car, on foot, in the air, or travelling by sea. When you look to purchase a suitcase or even a television, the concepts present in this skill are pondered to determine the right fit for us.