What different kinds of triangles are there?

Scalene, isosceles, equilateral, acute, right, obtuse.

In a right triangle, can two sides be the same length?

Yes, the base and height can be congruent.

In a right triangle, can three sides be the same length?

No, if all three sides were the same length, the triangle would only have 60° angles.

What are the sides of a right triangle called?

Base, height, and hypotenuse (it’s okay if students don’t name the “hypotenuse” yet).

Which side of a triangle is the base and which side is the height?

It depends on your point of view. The base is generally the bottom and the height is at a right angle to it, but the triangle can be rotated so that the side names are swapped.

When can you use the Pythagorean Theorem?

When I know two side lengths in a right triangle, and I want to calculate the missing side length.

How is squaring, as in a<sup>2</sup>, different from doubling, as in 2a?

a2=a⋅a, whereas 2a=a+a. For example, 32=9 and 3×2=6.

How would you find the length of the longest side of a triangle if the shorter two sides measure 6 units and 8 units?

Substitute these values into the Pythagorean Theorem to get 62+82=c2. Simplify to 100=c2, and then take the square root to get 10 units.

How would your calculations change if one leg measures 6 units and the hypotenuse measures 8 units?

I can still substitute 6 for a, but now I substitute 8 for c instead of b. The formula 62+b2=82 yields 36+b2=64, or b2=28. The exact value is square root of 28, which is about 5.3 units, rounded to the nearest tenth.

How can the Pythagorean Theorem help you find a distance on the coordinate plane?

I draw a triangle with the hypotenuse along the diagonal line that I want to measure. I count the lengths of the legs and substitute them for a and b in the Pythagorean Theorem. Then I can solve for c to find its length.

Pythagorean Theorem Math Video for Kids

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WHAT IS THE PYTHAGOREAN THEOREM?

The Pythagorean Theorem is a formula that relates the lengths of the legs and the length of the hypotenuse in a right triangle. The formula is a²+b²=c². You can use the formula to find a missing side length. You can also use the formula to see if a given set of side lengths could form a right triangle.

To better understand the Pythagorean Theorem…

WHAT IS THE PYTHAGOREAN THEOREM?. The Pythagorean Theorem is a formula that relates the lengths of the legs and the length of the hypotenuse in a right triangle. The formula is a2+b2=c2. You can use the formula to find a missing side length. You can also use the formula to see if a given set of side lengths could form a right triangle. To better understand the Pythagorean Theorem…

LET’S BREAK IT DOWN!

Birthday Party

The Pythagorean Theorem is a formula that shows how the side lengths of a right triangle are related. The shorter two sides of the triangle are called legs. One leg has length a and the other leg has length b. The longest side is called the hypotenuse. It has length c. The formula is a²+b²=c². Find the length of c if a = 3 and b = 4. 3² + 4² = 9 + 16 = 25. So, c² = 25. That means c = 5. Try this one yourself: For a right triangle, find the length of the hypotenuse if a=5 and b=12.

Birthday Party The Pythagorean Theorem is a formula that shows how the side lengths of a right triangle are related. The shorter two sides of the triangle are called legs. One leg has length a and the other leg has length b. The longest side is called the hypotenuse. It has length c. The formula is a2+b2=c2. Find the length of c if a = 3 and b = 4. 32 + 42 = 9 + 16 = 25. So, c2 = 25. That means c = 5. Try this one yourself: For a right triangle, find the length of the hypotenuse if a=5 and b=12.

Skateboard Ramp

The bottom of a skateboard ramp is 100 centimeters long. The height of the ramp is 30 centimeters. What is the length of the ramp part? Substitute 30 and 100 into the formula and find the value of c. 30² + 100² = 900 + 10,000 = 10,900. Take the square root of 10,900 to find the value of c. The result is irrational, so round to give an approximate answer. c ≈ 104.4 centimeters. The ramp part of the skateboard ramp is approximately 104.4 centimeters long. Try this one yourself: For a right triangle, find the length of the hypotenuse if the base is 6 m and the height is 7 m. Use a calculator, and round your answer to the nearest tenth.

Skateboard Ramp The bottom of a skateboard ramp is 100 centimeters long. The height of the ramp is 30 centimeters. What is the length of the ramp part? Substitute 30 and 100 into the formula and find the value of c. 302 + 1002 = 900 + 10,000 = 10,900. Take the square root of 10,900 to find the value of c. The result is irrational, so round to give an approximate answer. c ≈ 104.4 centimeters. The ramp part of the skateboard ramp is approximately 104.4 centimeters long. Try this one yourself: For a right triangle, find the length of the hypotenuse if the base is 6 m and the height is 7 m. Use a calculator, and round your answer to the nearest tenth.

Dog Ramp

The design for a dog ramp says that the longest side is 5 feet long and the base is 4 feet long. What is the height of the ramp? Substitute those two lengths into the formula. You can substitute 4 for either a or b. a²+4²=5², so a²+16=25. Subtract 16 from both sides to isolate the a². a²=9. Since 3×3=9, a=3. The height of the dog ramp is 3 feet. Try this one yourself: One leg of a right triangle is 6 feet long. The hypotenuse is 10 feet long. What is the length of the other side?

Dog Ramp The design for a dog ramp says that the longest side is 5 feet long and the base is 4 feet long. What is the height of the ramp? Substitute those two lengths into the formula. You can substitute 4 for either a or b. a2+42=52, so a2+16=25. Subtract 16 from both sides to isolate the a2. a2=9. Since 3×3=9, a=3. The height of the dog ramp is 3 feet. Try this one yourself: One leg of a right triangle is 6 feet long. The hypotenuse is 10 feet long. What is the length of the other side?

Beehouse

If you have a right triangle, then the side lengths make this equation true: a²+b²=c². Another mathematical fact is that, if a²+b²=c², then the triangle is a right triangle. For example, three pieces of wood are cut to lengths of 8 centimeters, 15 centimeters, and 17 centimeters. Can these three lengths fit together to make a bee house in the shape of a right triangle? The longest side is 17, so set c=17. Set a=8 and b=15 for the legs. Squaring these numbers, we get a²=64, b²=225, and c²=289. Then, the sum 64+225 is 289. Since a²+b² has the same value as c², that means the three pieces of wood can form a right triangle. Try this one yourself: Can the lengths 7 centimeters, 24 centimeters, and 25 centimeters form a right triangle?

Beehouse If you have a right triangle, then the side lengths make this equation true: a2+b2=c2. Another mathematical fact is that, if a2+b2=c2, then the triangle is a right triangle. For example, three pieces of wood are cut to lengths of 8 centimeters, 15 centimeters, and 17 centimeters. Can these three lengths fit together to make a bee house in the shape of a right triangle? The longest side is 17, so set c=17. Set a=8 and b=15 for the legs. Squaring these numbers, we get a2=64, b2=225, and c2=289. Then, the sum 64+225 is 289. Since a2+b2 has the same value as c2, that means the three pieces of wood can form a right triangle. Try this one yourself: Can the lengths 7 centimeters, 24 centimeters, and 25 centimeters form a right triangle?

Pythagorean Theorem

Discussion Questions

Vocabulary

Reading Material

LET’S BREAK IT DOWN!

Birthday Party

Skateboard Ramp

Dog Ramp

Beehouse

Practice Word Problems

Practice Number Problems

Teacher Resources

Lesson Plan

Teacher Guide

Answer Key

Assessment

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