Wentworth's Plane and Solid Geometry · Book III, Exs. 312–330 · Book 4: Book III: Proportion. Similar Polygons

55. Problems of computation (1)

Goal: Your student can compute altitudes, medians and bisectors of a triangle from its sides, and solve numerical problems on similar triangles and right triangles.

⏱ About 45 minutes · 6 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to these pages
  • Pencil and paper
  • A calculator for square roots (optional)

What's in the book

Pages 180–182: the worked formulas of Exercises 312–315 (altitudes, medians, bisectors and the circumscribed radius in terms of the sides), then numerical exercises 316–330.

The lesson, step by step

  1. Warm up

    Ask: State the theorems for the square of the side opposite an acute and an obtuse angle.

  2. The worked formulas

    Read Ex. 312 together. It derives the altitude from the three sides, using s for half the perimeter. Your student need not memorize the algebra but should follow each step.

  3. Acute, right or obtuse?

    Ex. 317. If the sides of a triangle are 7, 9, and 12, is the angle opposite 12 right, acute, or obtuse ?

    Compare 12² with 7² + 9². Do Ex. 316 and 318 the same way.

  4. Word problems

    Ex. 323. A tree casts a shadow 90 feet long, when a post 6 feet high casts a shadow 4 feet long. How high is the tree ?

    Ex. 327. How long a ladder is required to reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the house ?

  5. More

    Your student does three or four of Ex. 319–330, drawing a sketch for each.

  6. Wrap up

    Go over answers. Ask your student to explain one solution in words.

Answers to the book's problems

Ex. 316: right. Ex. 317: obtuse. Ex. 318: acute. Ex. 319: projections about 4.44 in and 9.98 in; distance about 6.66 in (exactly 24/√13). Ex. 320: altitudes about 8.71, 5.81, 4.36 in (on the sides 6, 9, 12); medians about 10.17, 8.35, 4.74 in; bisectors about 9.96, 7.35, 4.41 in; radius about 6.20 in. Ex. 321: 12 in. Ex. 322: the side 16 is cut into 48/7 and 64/7; the side 12 into 4.32 and 7.68; the side 9 into 27/7 and 36/7. Ex. 323: 135 ft. Ex. 324: bh/(a − b) and ah/(a − b). Ex. 325: 30 and 35. Ex. 326: 36 ft. Ex. 327: 26 ft. Ex. 328: (a/2)√3. Ex. 329: 2h/√3, or (2h/3)√3. Ex. 330: 20 in and 16 in.

Worked out for this site from the scan. If a number in your copy differs, trust the book.

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