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Aufgabe

Lösungsschritte

1. Understand the question

   ving for a missing side.
   -overlapping triangles and creating equ
   -altitude problems
   -area and perimeter
   additional review topic:
   triangle inequality
   trigonometry
   part 1: similar triangle definitions
   in the accompanying diagram of right triangle abc,
   altitude \\(\\overline{bd}\\) is drawn to hypotenuse \\(\\overline{ac}\\)
   which statement must always be true?

   1. \\(\\frac{ad}{ab} = \\frac{bc}{ac}\\)
   2. \\(\\frac{ad}{ab} = \\frac{ab}{ac}\\)
   3. \\(\\frac{bd}{bc} = \\frac{ab}{ad}\\)
   4. \\(\\frac{ab}{bc} = \\frac{bd}{ac}\\)
2. Explanation

   Step1: Identify Similar Triangles

   In right triangle \( ABC \) with altitude \( BD \) to hypotenuse \( AC \), we have three similar triangles: \( \triangle ABC \sim \triangle ADB \sim \triangle BDC \) (by AA similarity, as all right triangles and share a common angle).

   Step2: Corresponding Sides of Similar Triangles

   For similar triangles \( \triangle ADB \) and \( \triangle ABC \), the ratios of corresponding sides are equal.

   • In \( \triangle ADB \), sides are \( AD \) (adjacent to \( \angle A \)), \( AB \) (hypotenuse), and \( BD \) (opposite to \( \angle A \)).
   • In \( \triangle ABC \), sides are \( AB \) (adjacent to \( \angle A \)), \( AC \) (hypotenuse), and \( BC \) (opposite to \( \angle A \)).

   So, the ratio of \( AD \) (from \( \triangle ADB \)) to \( AB \) (from \( \triangle ADB \)) should equal the ratio of \( AB \) (from \( \triangle ABC \)) to \( AC \) (from \( \triangle ABC \)) because they are corresponding sides of similar triangles. Mathematically, this is \( \frac{AD}{AB} = \frac{AB}{AC} \).

   Step3: Analyze Other Options

   • Option 1: \( \frac{AD}{AB} = \frac{BC}{AC} \) – \( BC \) and \( AD \) are not corresponding sides, so this is incorrect.
   • Option 3: \( \frac{BD}{BC} = \frac{AB}{AD} \) – Sides do not correspond, so incorrect.
   • Option 4: \( \frac{AB}{BC} = \frac{BD}{AC} \) – Sides do not correspond, so incorrect.
3. Final answer
   1. \( \boldsymbol{\frac{AD}{AB} = \frac{AB}{AC}} \)

Antwort

Question Analysis

OCR Text

ving for a missing side.
-overlapping triangles and creating equ
-altitude problems
-area and perimeter
additional review topic:
triangle inequality
trigonometry
part 1: similar triangle definitions
in the accompanying diagram of right triangle abc,
altitude \\(\\overline{bd}\\) is drawn to hypotenuse \\(\\overline{ac}\\)
which statement must always be true?
1) \\(\\frac{ad}{ab} = \\frac{bc}{ac}\\)
2) \\(\\frac{ad}{ab} = \\frac{ab}{ac}\\)
3) \\(\\frac{bd}{bc} = \\frac{ab}{ad}\\)
4) \\(\\frac{ab}{bc} = \\frac{bd}{ac}\\)

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