Week 06 · Similar Triangles and Proof
Shop Day
Shop Day: the tallest thing in the yard
// Shadows, side-splitters and the altitude
⏱ about 20 min
Shop Day: The Tallest Thing in the Yard
Saturday afternoon the crew brings the meter stick to Comet's yard. "The tallest thing we can find," Comet says.
Wren points at the tall post holding the clothesline. "That one. What do you notice about its shadow?"
Comet's cousin paces it. "Long. About 2.4 meters." The stick's shadow is 0.6.
Nova projects the two triangles on the grass. "Would you like a hint? Divide the long shadow by the short one."
"Four. So the post is 4 meters," the cousin says.
"Check it another way," Wren says. "Measure the shadow again in ten minutes and see if you get the same height."
"Your turn to run it," Comet says.
- Two shadow triangles at one moment are similar by AA. Stick to shadow equals object to shadow.
- A line parallel to one side of a triangle splits the other two sides proportionally, and the converse holds.
- The altitude to the hypotenuse makes two triangles similar to the whole, which proves a² + b² = c².
- Congruence criteria (SSS, SAS, ASA) fix size and shape. Similarity criteria (AA, SSS for similarity) fix shape.
◇ MEASURE THE TALLEST THING IN THE YARD
Pick something tall with a shadow on flat ground: a post, a pole or a fence. Never climb it.
Stand the meter stick upright. One person marks both shadow tips at the same moment; another measures.
Divide the long shadow by the stick's shadow and multiply by the stick's height.
Wait ten minutes and repeat. Did the height come out the same? Then try a second tall object.
Stand the meter stick upright. One person marks both shadow tips at the same moment; another measures.
Divide the long shadow by the stick's shadow and multiply by the stick's height.
Wait ten minutes and repeat. Did the height come out the same? Then try a second tall object.
For a grown-up
Hold the stick upright and keep everyone looking down at the shadows, never up at the Sun.
Keep the measuring away from the road, and nobody climbs anything to check the answer.
If two heights disagree, ask whether both shadows were measured at exactly the same moment.
Keep the measuring away from the road, and nobody climbs anything to check the answer.
If two heights disagree, ask whether both shadows were measured at exactly the same moment.
SHOP DAY CHECK
- Read the question.
- Tap your answer.
The stick's shadow is 0.6 m and the post's shadow is 2.4 m. How tall is the post, in meters?
At the same moment the fence casts a 1.2 m shadow. How tall is the fence, in meters?
A guy line runs from the top of a 4 m post to the ground 3 m from its base. How long is the line, in meters?Ten minutes later the stick's shadow is 0.8 m and the post's is 3.2 m. How tall is the post now, in meters? Type the number.
Divide 3.2 by 0.8, then multiply by 1 meter.
WHY THIS EXERCISEThe same height from a different pair of shadows is the check that the method works.
Draw the post and the stick with their shadows, both triangles labeled with the family's numbers.