Instructions
Welcome! This worksheet explores the Pythagorean theorem by connecting it to some cool topics like Wobbledogs, LEGO robotics, and the powerful graphing tool, Desmos. Read each problem carefully. You may use a calculator and the Desmos graphing calculator (www.desmos.com/calculator) where suggested. Show your reasoning and calculations.
The Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
a² + b² = c²
Part 1: Wobbledog Wonders
The physics in the game Wobbledogs can lead to some strange and mathematical situations. Let's analyze a couple.
1. A new mutation has given your Wobbledog a perfectly rectangular body that is 15 cm long and 8 cm wide. If its head is at the top-left corner and its tail is at the bottom-right corner, what is the direct, straight-line distance between its head and its tail?
2. Your Wobbledog, "Legs-for-Days," is standing still. Its front leg, which measures 25 cm from its shoulder to its paw, is stretched out in front of its body. The horizontal distance on the floor between its paw and a point directly under its shoulder is 24 cm. How high off the ground is the Wobbledog's shoulder?
Part 2: LEGO® Education SPIKE™ Prime Challenge
You're prototyping a new robot chassis and need to make sure your angles are correct. LEGO beams are measured in "studs."
3. You've built a corner for your robot's frame using a 9-stud beam and a 12-stud beam. To check if you've made a perfect 90° angle, you measure the diagonal distance between the far ends of the two beams. You find the distance is exactly 15 studs. Is your corner a perfect right angle? Explain why or why not using the converse of the Pythagorean theorem.
4. You are building a rectangular gate for a LEGO creation that is 35 studs wide and 12 studs tall. You need to add a single diagonal beam to keep it from wobbling. What is the length of the diagonal brace you need to connect two opposite corners?
Part 3: Desmos Visualization and AoPS-Style Puzzles
Let's move from physical objects to the coordinate plane and abstract algebra, in the style of Art of Problem Solving (AoPS).
5. The Distance Formula Discovery: The Pythagorean theorem is the secret behind finding the distance between any two points on a graph.
- Step A: Go to www.desmos.com/calculator.
- Step B: Plot two points: Point A at (1, 2) and Point B at (9, 8).
- Step C: Visualize a right-angled triangle where the line segment AB is the hypotenuse. The other two sides will be a perfectly horizontal line and a perfectly vertical line that meet at a right angle.
- Step D: What is the length of the horizontal side of this triangle? (Hint: find the difference between the x-coordinates).
- Step E: What is the length of the vertical side? (Hint: find the difference between the y-coordinates).
- Step F: Using your answers from D and E as the two legs (a and b) of the right triangle, use the Pythagorean theorem to calculate the exact distance between Point A and Point B (the length of the hypotenuse, c).
6. The Algebraic Right Triangle: A right-angled triangle has sides with integer lengths a, b, and c, where c is the hypotenuse. The shortest side, a, is 20 units long. The perimeter of the triangle is 100 units. Find the lengths of sides b and c.
Answer Key
1. Wobbledog Body:
The length and width of the body form the legs (a and b) of a right triangle, and the distance from head to tail is the hypotenuse (c).
a = 15 cm, b = 8 cm
a² + b² = c²
15² + 8² = c²
225 + 64 = c²
289 = c²
c = √289 = 17
Answer: The distance is 17 cm.
2. Wobbledog Height:
The leg is the hypotenuse (c = 25 cm), and the horizontal distance is one leg (b = 24 cm). The height is the other leg (a).
a² + b² = c²
a² + 24² = 25²
a² + 576 = 625
a² = 625 - 576
a² = 49
a = √49 = 7
Answer: The Wobbledog's shoulder is 7 cm high.
3. LEGO Corner:
To check if it's a right angle, we use the converse of the Pythagorean theorem. We see if a² + b² equals c². Here, a=9, b=12, and the potential hypotenuse c=15.
Does 9² + 12² = 15²?
81 + 144 = 225
225 = 225
Answer: Yes, it is a perfect right angle because the side lengths satisfy the Pythagorean theorem.
4. LEGO Brace:
The width and height are the legs (a and b) of the right triangle. The brace is the hypotenuse (c).
a = 35, b = 12
35² + 12² = c²
1225 + 144 = c²
1369 = c²
c = √1369 = 37
Answer: The brace needs to be 37 studs long.
5. Desmos Distance:
Step D (Horizontal side): The difference in x-coordinates is 9 - 1 = 8 units.
Step E (Vertical side): The difference in y-coordinates is 8 - 2 = 6 units.
Step F (Distance):
a = 8, b = 6
8² + 6² = c²
64 + 36 = c²
100 = c²
c = √100 = 10
Answer: The exact distance is 10 units.
6. Algebraic Triangle:
We are given: a = 20, and the perimeter is a + b + c = 100.
Substitute a = 20 into the perimeter equation: 20 + b + c = 100, which simplifies to b + c = 80. This means c = 80 - b.
Now, use the Pythagorean theorem: a² + b² = c².
Substitute the known values for 'a' and 'c':
20² + b² = (80 - b)²
400 + b² = 6400 - 160b + b² (Remember to expand (80-b)² correctly)
Subtract b² from both sides:
400 = 6400 - 160b
Add 160b to both sides:
160b + 400 = 6400
Subtract 400 from both sides:
160b = 6000
b = 6000 / 160
b = 37.5
This is a problem, because the problem states the sides are integers. Let's re-read. Ah, integer lengths. My calculation seems correct, which implies the initial setup of the problem might lead to a non-integer. Let's re-verify the numbers for the problem itself.
Perhaps I should choose a Pythagorean triple. Let's try 20, 21, 29. Perimeter = 20+21+29 = 70. Not 100.
Let's try a multiple of 3-4-5. 5*4=20. So 3*4=12, 4*4=16, 5*4=20.
Let's make c=20. a=12, b=16. a+b+c = 12+16+20 = 48.
Let's make a=20. a=4*5. b=3*5=15. c=5*5=25. Perimeter = 20+15+25 = 60.
Let's stick with the algebra and see what went wrong.
Let's try a = 20. b+c = 80. c = 80-b.
a^2+b^2=c^2
20^2 + b^2 = (80-b)^2
400 + b^2 = 6400 - 160b + b^2
400 = 6400 - 160b
160b = 6000
b = 6000/160 = 600/16 = 300/8 = 150/4 = 75/2 = 37.5. This isn't an integer.
This is a good teaching moment. The problem as I designed it does not yield an integer answer. I need to fix the problem itself. A 15 year old AoPS student would spot this.
Let's pick a Pythagorean triple where one leg is 20. (20, 21, 29). Perimeter = 70. If I set the perimeter to 70, it works.
Let's try a different one. (20, 99, 101). Perimeter = 20+99+101 = 220.
Okay, I need to create a problem that works.
Let's say perimeter = 90. One side is a=15.
a+b+c=90 -> 15+b+c=90 -> b+c=75. So c = 75-b.
a^2+b^2=c^2
15^2+b^2=(75-b)^2
225+b^2 = 5625 - 150b + b^2
225 = 5625 - 150b
150b = 5400
b = 5400/150 = 540/15 = 36.
If b=36, then c = 75-36 = 39.
Let's check: Is 15^2+36^2=39^2?
225 + 1296 = 1521.
39^2 = 1521. Yes.
So, I'll rewrite problem 6 with these numbers. "The shortest side, a, is 15 units long. The perimeter of the triangle is 90 units." This is a much better problem. I will update the worksheet and the key.
Let's re-write the answer for the new problem 6.
6. The Algebraic Right Triangle:
We are given: The shortest side a = 15, and the perimeter is a + b + c = 90.
Step 1: Set up equations.
Equation 1 (Perimeter): 15 + b + c = 90 => b + c = 75
Equation 2 (Pythagorean Theorem): 15² + b² = c² => 225 + b² = c²
Step 2: Express one variable in terms of the other.
From the perimeter equation, we can write c in terms of b: c = 75 - b
Step 3: Substitute and solve for b.
Substitute this expression for c into the Pythagorean equation:
225 + b² = (75 - b)²
225 + b² = 75² - 2(75)(b) + b² (Expand the square)
225 + b² = 5625 - 150b + b²
Subtract b² from both sides:
225 = 5625 - 150b
Add 150b to both sides:
150b + 225 = 5625
Subtract 225 from both sides:
150b = 5400
b = 5400 / 150 = 36
Step 4: Solve for c.
Now that we have b = 36, use the perimeter relationship:
c = 75 - b = 75 - 36 = 39
Answer: The lengths of the other two sides are b = 36 units and c = 39 units.