Math Problem Statement

Pada gambar di bawah, panjang OC = 20 cm dan CE = 8 cm. Panjang tali busur AB adalah...

Solution

To solve this problem, let's analyze the geometry in the circle and apply appropriate calculations.

Given Information:

  • OC=20cmOC = 20 \, \text{cm}
  • CE=8cmCE = 8 \, \text{cm}
  • We need to find the length of the chord ABAB.

Solution Approach:

  1. Determine the Radius rr: Since OO is the center of the circle and OCOC is a radius, we have: r=OC=20cmr = OC = 20 \, \text{cm}

  2. Identify Triangle OECOEC: Triangle OECOEC is a right triangle with:

    • OEOE as one leg,
    • CE=8cmCE = 8 \, \text{cm} as the other leg,
    • OC=20cmOC = 20 \, \text{cm} as the hypotenuse.
  3. Calculate OEOE Using the Pythagorean Theorem: OE2+CE2=OC2OE^2 + CE^2 = OC^2 Substituting the values: OE2+82=202OE^2 + 8^2 = 20^2 OE2+64=400OE^2 + 64 = 400 OE2=336OE^2 = 336 OE=336=42118.33cmOE = \sqrt{336} = 4\sqrt{21} \approx 18.33 \, \text{cm}

  4. Determine ABAB: Since ABAB is a chord perpendicular to OEOE at EE, and EE is the midpoint of ABAB, we can calculate ABAB using the formula for a chord: AB=2OC2OE2AB = 2 \sqrt{OC^2 - OE^2} Substituting the values: AB=2202(421)2AB = 2 \sqrt{20^2 - (4\sqrt{21})^2} AB=2400336=264=2×8=16cmAB = 2 \sqrt{400 - 336} = 2 \sqrt{64} = 2 \times 8 = 16 \, \text{cm}

Answer:

The length of chord ABAB is 16cm\boxed{16 \, \text{cm}}, which corresponds to option A. 16 cm.


Do you want more details or have any questions?

Here are five related questions to explore further:

  1. How do we calculate the radius of a circle if given only a chord and a perpendicular distance from the center?
  2. What other methods can we use to find the length of a chord in a circle?
  3. How would the length of ABAB change if the radius of the circle was different?
  4. What is the significance of the Pythagorean theorem in solving circle problems?
  5. Can this problem be solved using trigonometric methods?

Tip: For problems involving chords in a circle, drawing perpendiculars from the center can often simplify calculations significantly.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Chords

Formulas

Pythagorean theorem: a^2 + b^2 = c^2
Chord length formula: AB = 2 * sqrt(OC^2 - OE^2)

Theorems

Pythagorean theorem

Suitable Grade Level

Grades 10-12