Advertisements
Advertisements
प्रश्न
In ∆ABC, E is the mid-point of the median AD, and BE produced meets side AC at point Q.
Show that BE: EQ = 3: 1.
Advertisements
उत्तर
Construction: Draw DX || BQ
In ΔBCQ and ΔDCX,
∠BCQ = ∠DCX ...(Common)
∠BQC = ∠DXC ...(Corresponding angles)
So, ΔBCQ ∼ ΔDCX ....(AA Similarity criterion)
⇒ `"BQ"/"DX" = "BC"/"DC" = "CQ"/"CX"` ...(Corresponding sides are proportional.)
⇒ `"BQ"/"DX" = "2CD"/"CD"` ...(D is the mid-point of BC)
⇒ `"BQ"/"DX" = 2` ...(i)
Similarly, ΔAEQ ∼ ΔADX,
⇒ `"EQ"/"DX" = "AE"/"ED" = 1/2` ...(E is the mid-point of AD)
That is `"EQ"/"DX" = 1/2` ...(ii)
Dividing (i) by (ii), We get
⇒ `"BQ"/"EQ" = 4`
⇒ BE + EQ = 4EQ
⇒ BE = 3EQ
⇒ `"BQ"/"EQ" = 3/1`
APPEARS IN
संबंधित प्रश्न
ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively
intersecting at P, Q and R. Prove that the perimeter of ΔPQR is double the perimeter of
ΔABC
In triangle ABC, M is mid-point of AB and a straight line through M and parallel to BC cuts AC in N. Find the lengths of AN and MN if Bc = 7 cm and Ac = 5 cm.
The side AC of a triangle ABC is produced to point E so that CE = AC. D is the mid-point of BC and ED produced meets AB at F. Lines through D and C are drawn parallel to AB which meet AC at point P and EF at point R respectively.
Prove that:
- 3DF = EF
- 4CR = AB
In ΔABC, D, E, F are the midpoints of BC, CA and AB respectively. Find ∠FDB if ∠ACB = 115°.
In the given figure, ABCD is a trapezium. P and Q are the midpoints of non-parallel side AD and BC respectively. Find: DC, if AB = 20 cm and PQ = 14 cm
In ΔABC, P is the mid-point of BC. A line through P and parallel to CA meets AB at point Q, and a line through Q and parallel to BC meets median AP at point R. Prove that: AP = 2AR
In ΔABC, P is the mid-point of BC. A line through P and parallel to CA meets AB at point Q, and a line through Q and parallel to BC meets median AP at point R. Prove that: BC = 4QR
ABCD is a parallelogram.E is the mid-point of CD and P is a point on AC such that PC = `(1)/(4)"AC"`. EP produced meets BC at F. Prove that: F is the mid-point of BC.
D and E are the mid-points of the sides AB and AC of ∆ABC and O is any point on side BC. O is joined to A. If P and Q are the mid-points of OB and OC respectively, then DEQP is ______.
Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.
