3D Applications of Pythagorean Theorem Warmup

Front

What is the Pythagorean Theorem?

Back

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It can be expressed as: c² = a² + b².

Front

How do you calculate the length of a ladder leaning against a wall?

Back

Use the Pythagorean Theorem. If the height from the ground to the point where the ladder touches the wall is 'a' and the distance from the wall to the base of the ladder is 'b', then the length of the ladder 'c' can be calculated as: c = √(a² + b²).

Front

If a ladder reaches 12 feet high and is placed 9 feet from the wall, what is the length of the ladder?

Back

Using the Pythagorean Theorem: c = √(12² + 9²) = √(144 + 81) = √225 = 15 feet.

Front

What is the diagonal length of a rectangle with sides 6 inches and 8 inches?

Back

Using the Pythagorean Theorem: c = √(6² + 8²) = √(36 + 64) = √100 = 10 inches.

Front

In 3D modeling, why is the Pythagorean Theorem important?

Back

It is used to measure distances between objects, calculate texture dimensions, and determine the correct scale of models, ensuring accuracy in virtual environments.

Front

What is the distance from the base of a house to place a 15' ladder to reach a 12' wall?

Back

Using the Pythagorean Theorem: b = √(15² - 12²) = √(225 - 144) = √81 = 9 feet.

Front

What is the formula for finding the distance between two points in 3D space?

Back

The distance formula in 3D is: d = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²).

Front

If the lengths of two sides of a right triangle are 5 and 12, what is the length of the hypotenuse?

Back

Using the Pythagorean Theorem: c = √(5² + 12²) = √(25 + 144) = √169 = 13.

Front

What is the relationship between the sides of a 30-60-90 triangle?

Back

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2, where the shortest side is opposite the 30-degree angle.

Front

How do you find the height of a right triangle if you know the base and hypotenuse?

Back

Use the Pythagorean Theorem: a = √(c² - b²), where 'c' is the hypotenuse and 'b' is the base.