Q9 of 56 Page 6

In a triangle ABC, XY || AC divides the triangle into two parts equal in areas. Determine .

Given: ABC is a triangle with XY || AC divides the triangle into two parts equal in areas.
To find:
Proof:
ar ΔBXY = ar trap. XYCA (Given) ∴ ar ΔBXY = ar ΔABC
In ΔBXY andBAC,
∠BXY = ∠BAC (Corresponding angles)
∠BYX = ∠BCA (Corresponding angles)
ΔBXY ∼ ΔBAC (AA similarity)
∴ = (Areas of similar triangle)
∴ =
∴ =
∴ AB – BX = √ 2 BX – BX
∴ AX = (√ 2 – 1)BX
= =.

More from this chapter

All 56 →
7 One angle of a triangle is equal to one angle of another triangle and the bisectors of these equal angles divide the opposite sides in the same ratio. Prove that the triangles are similar.
                                
8 The bisector of interior angle A of a triangle AB meets BC in D and the bisector of exterior angle A meets BC produced in E. Prove that
                                         10 If two sides and a median bisecting the third side of a Δ are respectively proportional to the corresponding sides and the median of another triangle, then prove that the two triangles are similar. 
                        11 Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same sides of BC. If AC and DB intersect at P, prove that AP × PC  = BP × PD.