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13-Foot Ladder Geometry Problem

13-Foot Ladder Geometry Problem
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2015-11-03 04:32
VectorTrigonometryCoordinate GeometryGeometric Construction
In collections CCSS Support Math II Unit 2
This example demonstrates a right triangle geometry problem involving a 13-foot ladder leaning against a building with its base 5 feet away. Key math concepts include the Pythagorean theorem, trigonometric functions (sine, cosine, tangent), and properties of right triangles. GeoGebra commands and objects used are Segment, Vector, LaTeX, and If to construct and analyze the position changes of the ladder's top. Suitable for high school students learning analytical geometry, trigonometric applications, and geometric constructions.

GeoGebra Commands & Parameters

  1. Segment((0, 0), (0, 1)) → i
  2. Segment(A_0, B_0) → Scaleline
  3. Sequence(Segment[A_0 + (i, 0), A_0 + (i, (-0.5))], i, 0, 10) → list1
  4. Sequence(Segment[A_0 + (i - 0.5, 0), A_0 + (i - 0.5, (-0.4))], i, 1, 10) → list2
  5. Sequence(Sequence[Segment[A_0 + ((0.5 * i) + (0.1 * j), 0), A_0 + ((0.5 * i) + (0.1 * j), (-0.3))], j, 1, 4], i, 0, 19) → list3
  6. LaTeX("\text{0}") → Scalevalue0
  7. LaTeX("\text{1}") → Scalevalue1
  8. LaTeX("\text{2}") → Scalevalue2
  9. LaTeX("\text{3}") → Scalevalue3
  10. LaTeX("\text{4}") → Scalevalue4
  11. LaTeX("\text{5}") → Scalevalue5
  12. LaTeX("\text{6}") → Scalevalue6
  13. LaTeX("\text{7}") → Scalevalue7
  14. LaTeX("\text{8}") → Scalevalue8
  15. LaTeX("\text{9}") → Scalevalue9
  16. LaTeX("\text{10}") → Scalevalue10
  17. Segment(D_0, (x(D_0), 0)) → j
  18. Vector(C_0, D_0) → u
  19. LaTeX("\text{the ladder reach?}") → text3
  20. LaTeX("\text{Simplify the expression under the radical symbol.}") → text4
  21. LaTeX((("\text{\textcolor{#000099}{=" * sub1) * "}")) → text9
  22. LaTeX("\text{Find the square root of the simplified expression.}") → text11
  23. LaTeX("\text{Expression under the radical symbol}") → text17
  24. LaTeX("\text{Subtract.}") → text20
  25. LaTeX((("\text{Simplify \textcolor{#000099}{" * value1) * "²}.}")) → text18
  26. LaTeX((("\text{Simplify \textcolor{#000099}{" * value2) * "²}.}")) → text19
  27. LaTeX((((("\text{\textcolor{#000099}{" * value1) * "²-") * value2) * "²}")) → text6
  28. LaTeX((((("\text{\textcolor{#000099}{=" * sqr1) * "-") * value2) * "²}")) → text7
  29. LaTeX((((("\text{\textcolor{#000099}{=" * sqr1) * "-") * sqr2) * "}")) → text8
  30. LaTeX((((((("\text{The expression \textcolor{#000099}{" * value1) * "² - ") * value2) * "²} simplifies to \textcolor{#000099}{") * sub1) * "}.}")) → text10
  31. LaTeX((((("\text{A \textcolor{#000099}{" * value1) * "}-\textcolor{#000099}{foot} ladder leans against the side of a building. The base of the ladder is \textcolor{#000099}{") * value2) * "\; feet} from the building.}")) → text1
  32. LaTeX((((("\text{Risa calculates that the top of the ladder rests \textcolor{#000099}{\sqrt{" * value1) * "² −") * value2) * "²}} feet up the side of the building. How high does}")) → text2
  33. LaTeX("\text{\mathbf{radical\;expression} an expression containing a root, such as \textcolor{#000099}{⁵\sqrt{9}}}") → WordsText
  34. LaTeX((("\text{First value=" * value1) * "}")) → text14
  35. LaTeX((("\text{Second value=" * value2) * "}")) → text15
  36. LaTeX((((("\text{The expression under the radical symbol is \textcolor{#000099}{" * value1) * "² - ") * value2) * "²}. Follow the order of operations to simplify this expression.}")) → text5
  37. If(sub1 ≟ 144, LaTeX[(((((((((((("\text{The simplified expression, \textcolor{#000099}{" * sub1) * "}, is a perfect square because \textcolor{#000099}{") * sqrt1) * " \;•\; ") * sqrt1) * " = ") * sub1) * "}. Therefore, \textcolor{#000099}{\sqrt{") * sub1) * "}=") * sqrt1) * "}.}")], LaTeX[(((((("\text{The simplified expression, \textcolor{#000099}{" * sub1) * "}, is not a perfect square. Therefore,\textcolor{#000099}{\sqrt{") * sub1) * "}=") * sqrt1) * "}.}")]) → text13
  38. LaTeX((("\text{Therefore, the top of the ladder reaches \textcolor{#000099}{" * sqrt1) * "} feet up the side of the building.}")) → text12

AI Drawing Prompt Reference

Draw a right triangle with one leg of 5 feet and a hypotenuse of 13 feet. Use a 13-foot ladder as the hypotenuse, with the base of the ladder 5 feet away from the building. Calculate the height where the top of the ladder touches the building using the Pythagorean theorem, and indicate the ladder's tilt with arrows.
The prompt above was compiled from the original drawing commands and is for reference only.