Geometry Class: Altitude in Right Triangles
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Questions and Answers

When an altitude is drawn from the right angle of a right triangle to its hypotenuse, what is the relationship between the newly formed triangles and the original triangle?

  • The new triangles are similar to each other, and also similar to the original triangle. (correct)
  • The new triangles are only similar to each other but not to the original triangle.
  • The new triangles are congruent to the original triangle.
  • The new triangles are not related to the original triangle.

    • What does the geometric mean theorem state about the altitude drawn to the hypotenuse in a right triangle?

  • The altitude is the average of the two segments it creates on the hypotenuse.
  • The altitude is the product of the two segments it creates on the hypotenuse.
  • The altitude is the square root of the product of the two segments it creates on the hypotenuse. (correct)
  • The altitude is equal to the sum of the two segments it creates on the hypotenuse.

    • In a right triangle, if the altitude drawn to the hypotenuse creates segments of lengths 4 and 9 on the hypotenuse, what is the length of the altitude?

  • 6.5
  • 6 (correct)
  • 13
  • 36

    • Given a right triangle with legs of length 5 and 12, what is the length of the hypotenuse?

    <p>13 (D)</p> Signup and view all the answers

    If the area of a right triangle is calculated using the two legs as the base and height, how else can the area be determined?

    <p>Using the hypotenuse as the base and the altitude to it as the height. (B)</p> Signup and view all the answers

    A right triangle has legs of lengths 6 and 8. What is the area using these legs and what is the length of the hypotenuse?

    <p>Area is 24, and the hypotenuse is 10. (D)</p> Signup and view all the answers

    In a right triangle with legs 6 and 8, and an altitude drawn to the hypotenuse, what is the length of this altitude?

    <p>4.8 (A)</p> Signup and view all the answers

    Which of these formulas is correct, h being the altitude and a, b, and c being the sides of the triangle?

    <p>$h^2 = ab$ (A)</p> Signup and view all the answers

    Flashcards

    Altitude in a Right Triangle

    The perpendicular line from the right angle to the hypotenuse, creating two similar triangles.

    Similar Triangles

    Triangles that have the same shape but not necessarily the same size; corresponding angles are equal.

    Geometric Mean

    A mean calculated by taking the square root of the product of two numbers; for altitude, h = √(ab).

    Altitude Formula

    For a right triangle, the altitude h is calculated as h² = ab, where a and b are segments of the hypotenuse.

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    Pythagorean Theorem

    A relation in right triangles: a² + b² = c², connecting the lengths of the legs and hypotenuse.

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    Area of a Right Triangle

    The area can be calculated using Area = (1/2) * base * height, often with legs or hypotenuse.

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    Area using Altitude

    When the altitude is used, Area can be calculated as (1/2) * h * hypotenuse.

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    Segments of Hypotenuse

    The parts into which the hypotenuse is divided by the altitude in a right triangle.

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    Study Notes

    Altitude in a Right Triangle

    • The altitude from the right angle to the hypotenuse creates two similar triangles within the original right triangle.
    • These similar triangles share angles with the original triangle, facilitating relationships between side lengths.

    Relationships in Similar Triangles

    • Similar triangles' corresponding sides have a consistent ratio.
    • This ratio applies to the sides of the similar triangles formed by the altitude.
    • The altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse.

    Geometric Mean

    • If 'h' is the altitude from the right angle to the hypotenuse, with segments of the hypotenuse 'a' and 'b':
      • h² = ab
      • h = √(ab)

    Pythagorean Theorem Relation

    • The Pythagorean Theorem (a² + b² = c²) relates the sides of a right triangle.
    • This theorem, combined with the altitude relationship, allows calculation of the altitude's length without needing its explicit value.
    • The altitude connects different side lengths of a right-angled triangle.

    Area Formula

    • A right triangle's area can be calculated using either leg as the base and the other leg as the height, or the hypotenuse as the base and the altitude as the height.
      • Area = (1/2) * base * height
    • The altitude to the hypotenuse offers an alternative area calculation. This is useful when leg lengths aren't known directly, instead knowing the hypotenuse segments or altitude length assists with calculation.

    Example

    • In a right triangle with legs 6 and 8:
      • The hypotenuse (c) is √(6² + 8²) = 10.
      • An altitude from the right angle to the hypotenuse divides the hypotenuse into segments 'a' and 'b'.
      • The altitude (h) can be found using the geometric mean formula or through the area formula:
        • Area = (1/2) * a * b
        • Using the area formula and (1/2) * h * 10 and (1/2) * 6 * 8 :
        • h * 10 = 48, h= 4.8

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    Description

    Explore the properties of altitudes in right triangles, focusing on the relationships between the triangle's sides and the altitude. Learn how the altitude divides the triangle into similar triangles and discover how it relates to the segments of the hypotenuse through the geometric mean.

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