The Shadow Investigation
Imagine you are outside on the school field. You look up at a massive oak tree.
You want to estimate the exact height of that tree. But there's a problem: you don't have a ladder, you can't climb it, and your tape measure is only 3 metres long. How can you possibly measure it?
Think about what you DO have:
- You know your own height.
- You can easily measure your own shadow on the ground.
- You can easily measure the distance from yourself to the tree.
Because the sun is so far away, its rays hit the Earth at the exact same angle for both you and the tree. This creates two Similar Triangles. One is just a perfectly scaled-up version of the other!
The Discovery
Try interacting with the environment on the right. You can use the zoom buttons, and drag the Sun to change the time of day.
Notice that no matter where the sun is, if you stand inside the tree's shadow so your shadow tips perfectly align, the triangles are locked in proportion!
Put it into Practice
You are exactly 1.6m tall, and your shadow is perfectly 2m long. You stand in the tree's shadow so your shadow tips align. The distance from your feet to the tree is exactly 10m.
Step 1: The Giant Shadow
Calculate the TOTAL length of the tree's shadow (Your shadow + Distance to tree).
Step 2: The Scale Factor
How many times larger is the tree's shadow compared to your shadow? ($12 \div 2$)
Step 3: The Final Height
Multiply your height (1.6m) by the scale factor to find the true height of the tree!
The Scale Factor ($k$)
When two shapes are similar, one is essentially a "zoomed in" or "zoomed out" version of the other. The multiplier we use to zoom is called the Scale Factor ($k$).
How to use the Scale Factor:
- Find a pair: Look for corresponding sides where you know both lengths.
- Calculate $k$: Divide the Big Side by the Small Side. ($k = \frac{\text{Big}}{\text{Small}}$).
- Apply it: Multiply a small side by $k$ to find its matching big side. (Or divide a big side by $k$ to find the small side).
Interactive Scaling
Drag the red point on the large triangle to the right. Watch how the scale factor mathematically links the corresponding sides.
Mastery Quiz
Instructions: Type the final numeric answer only. Do not include units (like 'm' or 'cm'). Questions get progressively harder!