Pythagorean Theorem Worksheet for 15-Year-Olds — AoPS Pre-Algebra & Intro to Algebra with LEGO Spike Prime & Wobbledogs

Part 1: Wobbledog Genetics & Ratios

In Wobbledogs, a dog's genes determine its physical traits. Let's simplify this concept. Imagine a gene for wings. The dominant allele is for no wings (N), and the recessive allele is for wings (w). A dog needs two recessive alleles (ww) to grow wings. You decide to breed two dogs that both have the heterozygous genotype (Nw).

1. What is the ratio of dominant alleles to recessive alleles in a single parent dog with the genotype Nw?
2. The potential genotypes for the puppies are NN, Nw, Nw, and ww. If the two dogs have a litter of 8 puppies, based on statistical probability, how many puppies would you expect to have wings?

Part 2: Lego Spike Prime & The Pythagorean Theorem

You have built a robot using your Lego Education Spike Prime set. You program it to perform a precise maneuver from its starting point, Point A.

• The robot moves 9 cm forward.
• It then turns exactly 90° to its left.
• It then moves 12 cm forward, stopping at Point B.

What is the straight-line distance between the robot's starting point (Point A) and its final position (Point B)?

Part 3: Algebraic Expressions for Gut Flora

The gut flora of your Wobbledog determines its mutations. You are trying to achieve a specific mutation by feeding it a precise diet of Chicken Legs and Cocoons.

• Each Chicken Leg provides 5 points of "Flora X" and 2 points of "Flora Y".
• Each Cocoon provides 3 points of "Flora X" and 4 points of "Flora Y".

1. Let c represent the number of Chicken Legs and n represent the number of Cocoons. Write an algebraic expression that represents the total "Flora X" points your dog will gain.
2. Using a similar expression for "Flora Y", calculate the total Flora Y points if your dog eats 4 Chicken Legs and 6 Cocoons.

Part 4: The Robotics & Algebra Challenge

You are designing a stationary Lego Spike Prime robotic arm to retrieve a special seed for your Wobbledog. The base of the robotic arm is at the origin (0,0) on a grid. The seed is located at a point that is 16 inches to the right of the base and 30 inches forward from the base.

1. The arm must extend in a straight line from its base to the seed. What is the minimum length the arm must be to reach the seed? (Hint: The path of the arm is the hypotenuse of a right triangle.)
2. The operational efficiency (E) of your robot is calculated with the formula:
   E = (L² - 100) / p
   Where L is the arm length you just calculated, and p is the number of processors in your Spike Prime hub (which is 1). What is the operational efficiency (E) of your robot?

Part 1: Wobbledog Genetics & Ratios

1. The ratio of dominant (N) to recessive (w) alleles is 1:1.
2. The probability of a puppy having the 'ww' genotype is 1 in 4 (or 25%). For a litter of 8 puppies, you would expect 2 puppies to have wings (0.25 * 8 = 2).

Part 2: Lego Spike Prime & The Pythagorean Theorem

Using the Pythagorean theorem, a² + b² = c².

• a = 9 cm, b = 12 cm
• 9² + 12² = c²
• 81 + 144 = c²
• 225 = c²
• c = √225
• c = 15

The straight-line distance is 15 cm.

Part 3: Algebraic Expressions for Gut Flora

1. Total Flora X = 5c + 3n
2. The expression for Flora Y is 2c + 4n.
   2(4) + 4(6) = 8 + 24 = 32.
   The total Flora Y is 32 points.

Part 4: The Robotics & Algebra Challenge

1. Using the Pythagorean theorem, a² + b² = c².
   • a = 16 in, b = 30 in
   • 16² + 30² = L² (where L is the arm length)
   • 256 + 900 = L²
   • 1156 = L²
   • L = √1156 = 34
   The minimum arm length is 34 inches.
2. Using the efficiency formula E = (L² - 100) / p.
   • L = 34, p = 1
   • E = (34² - 100) / 1
   • E = (1156 - 100) / 1
   • E = 1056 / 1 = 1056
   The operational efficiency is 1056.