Instructions
Your Wobbledog is a brilliant, if wobbly, engineer! It has discovered a LEGO® Education Spike Prime kit and wants to build some amazing structures. However, its understanding of geometry is a bit shaky. Your job is to help it calculate the correct lengths for its creations using the Pythagorean theorem. Remember, for any right-angled triangle, the theorem states:
a2 + b2 = c2
Where 'a' and 'b' are the lengths of the two shorter sides (the legs) that meet at the right angle, and 'c' is the length of the longest side opposite the right angle (the hypotenuse).
Part 1: Building a Stable Wall
Your Wobbledog has built a rectangular wall that is 9 LEGO units high and 12 LEGO units long. To stop it from wobbling, it needs to add a diagonal support beam from one corner to the opposite corner. What is the length of the beam the Wobbledog needs to find? This beam will be the hypotenuse of the triangle.
Show your work here:
Length of the diagonal beam (c): _______________ units
Part 2: The Ramp to the Food Bowl
The food bowl is on a platform. The Wobbledog has a long LEGO beam to use as a ramp that is 34 units long. It places the base of the ramp 16 units away from the platform on the floor. How high up the platform will the ramp reach?
Hint: In this case, you know the hypotenuse (the ramp) and one of the legs (the distance on the floor). You need to find the other leg (the height).
Show your work here:
Height the ramp reaches (a): _______________ units
Part 3: Translating the Context
Your Wobbledog sees a delicious treat on a shelf. It wants to build a ramp to get it. Read the description of the situation and translate it into a Pythagorean theorem problem to find the shortest possible ramp length.
"The floor is perfectly flat and meets the wall at a 90-degree angle. The distance from the base of the wall directly to the spot under the treat is 21 units. The height from the floor straight up to the treat is 20 units."
1. Identify the lengths of the legs (a and b) and the hypotenuse (c) from the text.
- a = __________
- b = __________
- c = ? (This is the ramp length you need to find)
2. Now, use the Pythagorean theorem to calculate the required length of the ramp.
Show your work here:
Minimum ramp length (c): _______________ units
Part 4: Algebraic Blueprints
Your Wobbledog finds a strange blueprint for a "perfectly proportional" triangular support structure. The lengths of the sides are written as algebraic expressions, not numbers!
- Shortest leg (a): x
- Longer leg (b): x + 1
- Hypotenuse (c): x + 2
Your challenge is to figure out the actual lengths. First, set up an equation using the Pythagorean theorem with these expressions. This is how you "see the structure" in algebra!
A) Set up the equation:
( )2 + ( )2 = ( )2
B) Now, solve for x. You will need to expand the squared terms. Remember that (x + 1)2 = (x + 1)(x + 1). Simplify the equation and solve for x. (A negative length is impossible!)
Show your work here:
The value of x is: _______________
C) What are the actual lengths of the three sides of the triangular support?
- Shortest leg: _______________ units
- Longer leg: _______________ units
- Hypotenuse: _______________ units
Extension: Visualize with Desmos
You can use the free online graphing calculator, Desmos, to visualize these problems. For Part 3, try plotting the three corners of the triangle as coordinates: (0, 0), (21, 0), and (0, 20). You can see the right triangle instantly! You can also type distance((21,0),(0,20)) into Desmos to have it calculate the hypotenuse for you and check your answer.
Answer Key
Part 1: Building a Stable Wall
a = 9, b = 12
92 + 122 = c2
81 + 144 = c2
225 = c2
√225 = c
15 = c
Length of the diagonal beam (c): 15 units
Part 2: The Ramp to the Food Bowl
c = 34, b = 16
a2 + 162 = 342
a2 + 256 = 1156
a2 = 1156 - 256
a2 = 900
a = √900
a = 30
Height the ramp reaches (a): 30 units
Part 3: Translating the Context
1. Identify the lengths:
- a = 20
- b = 21
- c = ?
2. Calculate the ramp length:
202 + 212 = c2
400 + 441 = c2
841 = c2
√841 = c
29 = c
Minimum ramp length (c): 29 units
Part 4: Algebraic Blueprints
A) Set up the equation:
(x)2 + (x + 1)2 = (x + 2)2
B) Solve for x:
x2 + (x2 + 2x + 1) = (x2 + 4x + 4)
2x2 + 2x + 1 = x2 + 4x + 4
(Subtract x2, 4x, and 4 from both sides to set the equation to zero)
x2 - 2x - 3 = 0
(Factor the quadratic equation)
(x - 3)(x + 1) = 0
The possible solutions are x = 3 or x = -1. Since length cannot be negative, x must be 3.
The value of x is: 3
C) Actual lengths:
- Shortest leg (x): 3 units
- Longer leg (x + 1): 4 units
- Hypotenuse (x + 2): 5 units
(This is a classic 3-4-5 right triangle!)