Similar Overlapping Triangles Game/Worksheet

Similar Overlapping Triangles Game/Worksheet
Welcome to the Similar Overlapping Triangles Challenge! In this game, smaller triangles are nested perfectly inside larger ones, sharing lines and angles. Learn how to untangle these overlapping triangles to find missing lengths. Scroll down the page for a more detailed explanation.

 

 

How to Play

1. Analyze the Figure: On the screen, you will see a small blue triangle (△ ADE) nested perfectly inside a larger green triangle (△ ABC).
   

2. Find the Variable: Look closely at the side labels. Three of the dimensions will be given as numbers, and one dynamic dimension will be labeled as an unknown pink x.
   

3. Submit Your Answer: Calculate the exact value of x, type it into the input box, and click Calculate x (or press Enter).
   

4. Learn and Level Up: If you are correct, you earn 100 points! If you make a mistake, the game immediately displays a step-by-step algebraic breakdown showing you exactly how to solve it. Click Next Level to try a brand-new, randomly generated challenge.
   

How the Math Works
Because the inner line DE is parallel to the base BC, the small top triangle shares the exact same angles as the large outer triangle. This makes them similar triangles, meaning their matching sides are perfectly proportional to one another.

To solve for x, you must set up a ratio comparing the small triangle to the large triangle:

\(\frac{\text{Small Left Side}}{\text{Large Left Side}} = \frac{\text{Small Base}}{\text{Large Base}}\)

Beware the Trapped Segment (DB)The most common mistake students make is comparing the small top segment (AD) to the bottom piece (DB). The bottom piece is not a side of a triangle—it is just a fragment. To find the true side length of the large triangle, you must add the two pieces together:

Full Left Side (AB) = AD + DB

Example Breakdown
Imagine the game gives you these numbers:
Small top side (AD) = 3
Bottom segment (DB) = 6
Small base (DE) = 4
Large base (BC) = x

1. Find the full side: The full left side AB is 3 + 6 = 9.
   
2. Set up the proportion: Match the small parts to the large parts.
   

\(\frac{\text{Small Side } (AD)}{\text{Large Side } (AB)} = \frac{\text{Small Base } (DE)}{\text{Large Base } (BC)} \implies \frac{3}{9} = \frac{4}{x}\)

3. Cross-multiply and solve:
   

3 · x = 9 · 4
3x = 36 ⇒ x = 12

No matter which piece the game hides as x, just remember: always compare a complete triangle side to a complete triangle side.

Similar Triangles - Find the Missing Side

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