experts plshh give me the proof of all three theorems ,,, ??? faster i need it !!!
i mean theorem 9.1 , 9.2 and 9.3 ???
Theorem 9.2: Two triangles on the same base and between the same parallels are equal in area.
Given: Δ ABC and Δ PBC lie on the same base BC and between the same parallels l and m.
To prove: Area (ΔABC) = Area (ΔPBC)
Construction: Draw AD and PQ, the altitude drawn from A and P on BC respectively.
Proof :
AD = PQ [Perpendicular distance between two parallel line is always equal] .... (1)
Given, Area (ΔABC)
and Area (ΔPBC)
⇒ Area (ΔABC) = Area (ΔPBC)
[Hence proved]
Theorem 9.3: Two triangles having the same base (or equal bases) and equal areas lie between the same parallels
Given: ΔABC and ΔPBC lie on the same base BC. Area (ΔABC) = Area (ΔPBC). AD and PQ are the altitude drawn from A and P on BC respectively.
To prove: AP || BC
Proof :
Given, Area (ΔABC) = Area (ΔPBC)
⇒ AD = PQ
So, the altitudes AD and PQ drawn on side BC are equal. Therefore, the triangle must lie between the same parallel.
Thus, AP || BC (Distance between parallel lines is constant)
Hope you get it!!