Math Problem Statement

Solve for the length of segment a in a circle where two intersecting chords split the segments as: a and 4 cm on one chord, and 3 cm and 6 cm on the other.

Solution

The problem asks to find the length of segment aa, and the provided formula is:

ab=cdab = cd

Where two segments intersect inside the circle, the product of the lengths of the segments on one line equals the product of the lengths of the segments on the other line.

From the diagram, we can identify:

  • One segment is split into aa and 4cm4 \, \text{cm},
  • The other segment is split into 3cm3 \, \text{cm} and 6cm6 \, \text{cm}.

Using the formula ab=cdab = cd:

a×4=3×6a \times 4 = 3 \times 6

Now solve for aa:

a×4=18a \times 4 = 18

a=184=4.5cma = \frac{18}{4} = 4.5 \, \text{cm}

Thus, the length of segment aa is 4.5cm4.5 \, \text{cm}.

Would you like further details or have any questions?

Related Questions:

  1. How do we derive the formula ab=cdab = cd for intersecting chords in a circle?
  2. What happens when two segments intersect outside the circle?
  3. Can we apply the same formula if the segments intersect on the circumference of the circle?
  4. How does this formula relate to the power of a point theorem?
  5. What other geometric properties can we deduce from intersecting chords?

Tip:

When solving problems involving intersecting chords, always remember that the product of the parts of one chord equals the product of the parts of the other chord, no matter where the intersection occurs inside the circle.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Theorems
Chords
Algebra

Formulas

ab = cd

Theorems

Intersecting Chords Theorem

Suitable Grade Level

Grades 8-10