Chapter 16

Further Geometry

You can see a marker across a river, but you can't stretch a tape measure to it. What could you measure from your side to find out how far away it is? If you've finished the core Geometry course, Further Geometry offers optional investigations that start with familiar shapes and ask what a few clues can tell you. Sometimes deciding whether you have enough clues is part of the puzzle.

What You'll Explore

7 lessons.

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Advanced Some lessons are marked Advanced because they go beyond core Geometry. Skip them or explore them if you are curious.

1. 1
   Generating Pythagorean Triples Advanced. This lesson goes beyond core Geometry. You can skip it. →

   Done Skipped Revisit Clear ▾

   Three, four, and five are familiar side lengths for a right triangle, but how could you find more examples with whole number sides? You'll look for a pattern and ask when a triangle you find is just a larger version of one you already know.

2. 2
   Triangle Area with Sine Advanced. This lesson goes beyond core Geometry. You can skip it. →

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   Picture two sides of a triangle joined at a hinge, with the third side changing as the hinge opens. How much can the two fixed lengths and the angle between them tell you about the area, even when the angle opens past a right angle?

3. 3
   The Law of Sines Advanced. This lesson goes beyond core Geometry. You can skip it. →

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   Suppose someone gives you two side lengths and an angle that isn't between them. As you investigate how a triangle's sides and angles are related, ask what it would take to decide whether those three measurements describe a single triangle.

4. 4
   The Law of Cosines Advanced. This lesson goes beyond core Geometry. You can skip it. →

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   Two straight trails leave the same fork, and each leads to a lookout. If you know the distance from the fork to each lookout and the angle between the trails, how could you find the straight distance from one lookout to the other?

5. 5
   Heron's Formula Advanced. This lesson goes beyond core Geometry. You can skip it. →

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   A triangular garden has a fence along each side, and you know how long each stretch of fence is. Could you work out how much ground the garden covers from those three lengths alone, without measuring across the garden?

6. 6
   Coordinate Area Formulas Advanced. This lesson goes beyond core Geometry. You can skip it. →

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   Mark the corners of an oddly shaped park on a coordinate grid. If someone gave you only the list of coordinates, what could you learn about the park's area, and would it matter which way you went around its boundary?

7. 7
   Constructing Tangents from an External Point Advanced. This lesson goes beyond core Geometry. You can skip it. →

   Done Skipped Revisit Clear ▾

   From a point outside a circle, try sketching a line that just touches the circle. How could you locate the point of contact exactly with a compass and straightedge, and how would you check that you've found every line that works?