Q. 245.0( 3 Votes )
D is the mid-point of side BC of a Δ ABC. AD is bisected at the point E and BE produced cuts AC at the point X. Prove that BE: EX = 3 : 1.
Given: In ΔABC, D is the midpoint of BC and E is the midpoint of AD.
To prove: BE: EX = 3: 1
1.) If two corresponding angles of two triangles are equal, the triangles are said to be similar.
2.) Basic proportionality theorem:
If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides into same ratio.
Const: Through D, Draw DF||BX
In ΔEAX and Δ ADF
∠EAX = ∠ADF (Common)
∠AXE = ∠DAF (Corresponding angles)
Then, ΔEAX ~ Δ ADF
(Corresponding parts of similar triangle are proportion)
(AE = ED given)
⇒ DF = 2EX. ……………(i)
In ΔCDF and ΔCBX (By AA similarity)
SO, (Corresponding parts of similar triangle area proportion)
(BD = DC given)
⇒BE + EX = 2DF
⇒BE = EX = 4EX
⇒BE = 4EX – EX
⇒BE = 4EX – EX
⇒BE = 3EX
Rate this question :
In an isosceles triangle ABC, if AB = AC = 13 cm and the altitude from A on BC is 5 cm, find BC.RD Sharma - Mathematics
In a , AB = BC = CA = 2 a and . Prove that
(i) (ii) Area () =RD Sharma - Mathematics
The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its tip reach?RD Sharma - Mathematics
The sides of certain triangles are given below. Determine which of them are right triangles.
(i) a = 7 cm, b = 24 cm and c = 25 cm
(ii) a = 9 cm, b = 16 cm and c = 18 cm
(iii) a = 1.6 cm, b = 3.8 cm and c = 4 cm
(iv) a = 8 cm, b = 10 cm and c = 6 cmRD Sharma - Mathematics
Each side of a rhombus is 10 cm. If one of its diagonals is 16 cm find the length of the other diagonal.RD Sharma - Mathematics
Two poles of height 9 m and 14 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.RD Sharma - Mathematics
In an isosceles triangle ABC, AB = AC = 25 cm, BC = 14 cm. Calculate the altitude from A on BC.RD Sharma - Mathematics
ABCD is a square. F is the mid-point of AB. BE is one third of BC. If the area of ΔFBE = 108 cm2, find the length of AC.RD Sharma - Mathematics
is a right triangle right-angled at A and . Show that
(iii) (iv)RD Sharma - Mathematics
In an equilateral, , prove that .RD Sharma - Mathematics