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If a quadrilateral has sides of lengths 5 cm, 12 cm, 13 cm, and 14 cm, is it a c

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Question: If a quadrilateral has sides of lengths 5 cm, 12 cm, 13 cm, and 14 cm, is it a cyclic quadrilateral?

Options:

  1. Yes
  2. No
  3. Not enough information
  4. Only if it is a rectangle

Correct Answer: No

Solution: 

A quadrilateral is cyclic if the sum of opposite angles is 180 degrees. The given side lengths do not provide enough information to determine if it is cyclic.

If a quadrilateral has sides of lengths 5 cm, 12 cm, 13 cm, and 14 cm, is it a c

Practice Questions

Q1
If a quadrilateral has sides of lengths 5 cm, 12 cm, 13 cm, and 14 cm, is it a cyclic quadrilateral?
  1. Yes
  2. No
  3. Not enough information
  4. Only if it is a rectangle

Questions & Step-by-Step Solutions

If a quadrilateral has sides of lengths 5 cm, 12 cm, 13 cm, and 14 cm, is it a cyclic quadrilateral?
  • Step 1: Understand what a cyclic quadrilateral is. A cyclic quadrilateral is a four-sided figure where all its corners (angles) can be inscribed in a circle.
  • Step 2: Know that for a quadrilateral to be cyclic, the sum of the opposite angles must equal 180 degrees.
  • Step 3: Realize that the side lengths alone (5 cm, 12 cm, 13 cm, and 14 cm) do not give us information about the angles.
  • Step 4: Conclude that without knowing the angles, we cannot determine if the quadrilateral is cyclic.
  • Cyclic Quadrilateral – A quadrilateral is cyclic if its vertices can all lie on a single circle, which is equivalent to the condition that the sum of each pair of opposite angles is 180 degrees.
  • Ptolemy's Theorem – For a cyclic quadrilateral, the sum of the products of its two pairs of opposite sides is equal to the product of its diagonals.
  • Triangle Inequality Theorem – In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
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