1. **Problem Statement:** You are designing a circular park and need to apply theorems involving chords, arcs, secants, and tangents. 2. **Chord Theorem:** - Identify a chord, say chord AB. - Draw its perpendicular bisector, which is a line that cuts AB into two equal parts at a right angle. - This bisector passes through the center of the circle because the perpendicular bisector of any chord in a circle always passes through the center. 3. **Intersecting Chords Theorem:** - The intersecting chords are AB and CD, intersecting at point E inside the circle. - Given AE = 4 cm and EB = 6 cm, the product AE × EB = $4 \times 6 = 24$ cm². - If CE = 3 cm, then by the theorem, AE × EB = CE × ED, so $24 = 3 \times ED$. - Solving for ED gives $ED = \frac{24}{3} = 8$ cm. 4. **Tangent–Chord Angle Theorem:** - The tangent touches the circle at point P. - Draw a chord from P to another point on the circle, say point A. - The angle between the tangent at P and chord PA equals the angle subtended by chord PA in the alternate segment of the circle. **Final answers:** - The perpendicular bisector of chord AB passes through the center. - $AE \times EB = 24$ cm². - $ED = 8$ cm. - The tangent–chord angle theorem relates the angle between tangent and chord to the angle in the alternate segment.