2 equal cords AB and CD intersect at E.prove that arc AD=arcCB
To prove that arc AD is equal to arc CB, we need to show that the corresponding angles of these arcs are equal. Since the two cords AB and CD are equal in length and intersect at point E, we can use the properties of intersecting chords to prove the equality of the corresponding arcs.
Here's how you can prove it:
1. Draw a diagram: Start by drawing a circle and two equal chords AB and CD that intersect at point E within the circle.
2. Identify the angles: In the diagram, find the angles formed by the intersections of the chords with the circle. Let's call the angle formed by chord AB and arc AD as angle α, and the angle formed by chord CD and arc CB as angle β.
3. Use the intersecting chord theorem: According to the intersecting chord theorem, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. In this case, we can write:
AE * BE = CE * DE
This is because AB and CD are equal in length.
4. Analyze the angles: Now, consider the angles within the triangle ABE and CDE. Angle α is formed by chord AB and arc AD, while angle β is formed by chord CD and arc CB.
5. Apply the angle-angle similarity: Since triangle ABE and CDE have equal corresponding angles (right angles at E) and the same included angle (angle BAE = angle CDE), we can conclude that the triangles are similar by the angle-angle similarity criterion.
6. Use the similar triangles property: Because the triangles are similar, the ratio of any two corresponding sides is equal. Therefore, we have:
AE / DE = BE / CE
Rearranging the equation, we get:
AE * BE = CE * DE
This is the same equation we obtained from the intersecting chord theorem.
7. Conclusion: Since the angles α and β are formed by equal corresponding sides within similar triangles, we can conclude that angle α is equal to angle β. Hence, arc AD is equal to arc CB.
By following these steps and applying the properties of intersecting chords and similar triangles, you can prove that arc AD is equal to arc CB.