The altitude to the hypotenuse and the geometric mean
- Three triangles in one picture — 3 tasks
- The number that makes the areas equal — 3 tasks
- Three triangles of one shape — 3 tasks
- Reading the relations off the picture — 3 tasks
- Going Deeper — 6 tasks
- Practice — 24 tasks
- Summary — 2 tasks
- Cool-down — 3 tasks
Classroom activities
Things a class does rather than writes on. Each one is a finished PDF: print it, cut it up if it says so, and hand it round.
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Which Rule Finds the Length? The Altitude to the Hypotenuse — Cut & Glue Sort 25 minutes, including the cutting · 5 sheets
12 triangles, each with a segment drawn from one corner, two known lengths and one to find, and no numbers. Students decide which rule finds the length: the altitude rule, a leg rule, or neither, because a right-angle square is missing. The pieces hold the choices that matter: a leg asked when the two pieces are given, and a segment that only looks like the altitude.
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The Geometric Mean, Not the Average: The Altitude to the Hypotenuse — Answer Maze 10–15 minutes · 4 sheets
8 boxes on one page, all about one right triangle with the altitude drawn to its hypotenuse. Each box gives two lengths and asks for a third: the altitude, a piece of the hypotenuse, or a leg, with each rule run forwards and backwards. The wrong answers are the two costliest mistakes on this topic: the average in place of the geometric mean, and a leg paired with the wrong piece. The row of box numbers at the foot is the check.
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Is This Equation True? The Altitude to the Hypotenuse — Mystery Picture 25–35 minutes · 4 sheets
25 squares, each one equation written with the letters of one of three right triangles with the altitude drawn: fill the square if the equation is true, leave it white if it is not. The equations come as squares and as proportions, and the white squares are the mistakes students really make: the average for the geometric mean, a leg paired with the wrong piece, and a piece where the whole hypotenuse belongs. When every square is right, a picture appears.
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The Altitude Rule or a Leg Rule? The Altitude to the Hypotenuse — Scavenger Hunt a full class period · 13 sheets
16 cards around the room, each one right triangle with the altitude drawn to its hypotenuse. The cards give two lengths and ask for the altitude, a piece of the hypotenuse, or a leg. Taking the average instead of the geometric mean, using the two pieces where the whole hypotenuse belongs, or pairing a leg with the wrong piece lands on a card that sends the group back, and a clean run is 8 lines on the record sheet.