Chapter 9.1, Problem 18WE

To determine

To Draw: Two congruent circles with radii 6 each passing through the centre of other and to find length of their common chord.

Answer to Problem 18WE

  $6\sqrt{3}$

Explanation of Solution

Given information:

Radii of both the circles are 6 and they pass through the centre of each other.

Concept Use:

Perpendicular from center of circle to the chord, bisects the chord.

Calculation:

Two circles can be drawn as:

  

To find the length of the common chord $AB$ ,in triangle ADC.

  $\begin{array}{l}AD=\sqrt{A{C}^2+C{D}^2}\\ AD=\sqrt{A{C}^2+{\left(\frac{DE}{2}\right)}^2}\\ 6=\sqrt{A{C}^2+{\left(\frac{6}{2}\right)}^2}\\ 6^2=A{C}^2+{\left(\frac{6}{2}\right)}^2\\ 36=A{C}^2+9\\ A{C}^2=27\\ AC=\sqrt{27}\\ AC=3\sqrt{3}\end{array}$

Perpendicular to any chord bisects the chord therefore,

  $\begin{array}{l}AB=2AC\\ AB=2\times \left(3\sqrt{3}\right)\\ AB=6\sqrt{3}\end{array}$

Therefore, the answer is $6\sqrt{3}$ .