In Fig. 6.10, if AB||DC and AC and PQ intersect each other at the point O, prove that OA . CQ = OC . AP
Solution:
Given, AB||DC
AC and PQ intersect each other at point O.
We have to prove that OA . CQ = OC . AP
In △POA and △COQ,
The vertically opposite angles are equal
i.e., ∠POA = ∠COQ
Since AB||DC and PQ is transversal,
The alternate angles are equal.
∠APO = ∠CQO
AAA criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.
By AAA criterion, the third angle will be equal.
i.e., ∠PAO = ∠QCO
Therefore, the triangles POA and COQ are similar.
By the property of similar triangles,
The corresponding sides are proportional.
OA/OC = AP/CQ
On cross multiplication,
OA . CQ = AP . OC
Therefore, it is proved that OA . CQ = OC . AP.
✦ Try This: O is the point of intersection of the diagonals AC and BD of a trapezium ABCD with AB || DC. Through O, a line segment PQ is drawn parallel to AB meeting AD in P and BC in Q, prove that PO = QO
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6
NCERT Exemplar Class 10 Maths Exercise 6.3 Problem 5
In Fig. 6.10, if AB||DC and AC and PQ intersect each other at the point O, prove that OA . CQ = OC . AP
Summary:
In Fig. 6.10, if AB||DC and AC and PQ intersect each other at the point O, it is proven that OA . CQ = OC . AP
☛ Related Questions: