Advertisements
Advertisements
प्रश्न
D, E, and F are the mid-points of the sides AB, BC, and CA respectively of ΔABC. AE meets DF at O. P and Q are the mid-points of OB and OC respectively. Prove that DPQF is a parallelogram.
Advertisements
उत्तर
Given: △ABC, D, E, F are midpoints of AB, BC, AC respectively. AB and DF meet at O. P and Q are midpoints of OB and OC respectively.
To Prove: DPFQ is a parallelogram.
Proof:
In △ABC,
D is the mid-point of AB and F is the mid-point of AC
Hence, DF ∥ BC and DF = `1/2`BC ... (1) (Mid-point theorem)
In △OBC,
P is the mid-point of OB and Q is the mid-point of OC
Hence, PQ ∥ BC and PQ = `1/2`BC ... (2) (mid-point theorem)
thus, from (1) and (2)
DF ∥ PQ and DF = PQ ....(3)
Now, In △AOB,
D is the mid-point of AB and P is the mid-point of OB
Thus, DP ∥ AE and DP = `1/2`AE ....(4) (midpoint theorem)
Now, In △AOC,
F is the midpoint of AC and Q is the midpoint of OC
Thus, FQ ∥ AE and QF = `1/2`AE .....(5) (midpoint theorem)
thus, from (4) and (5)
DP ∥ FQ and DP = FQ .....(6)
DPFQ is a parallelogram ......(from (3) and (6))
Hence proved.
APPEARS IN
संबंधित प्रश्न
ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see the given figure). AC is a diagonal. Show that:
- SR || AC and SR = `1/2AC`
- PQ = SR
- PQRS is a parallelogram.
ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see the given figure). Show that F is the mid-point of BC.
Fill in the blank to make the following statement correct:
The triangle formed by joining the mid-points of the sides of a right triangle is
D, E, and F are the mid-points of the sides AB, BC and CA of an isosceles ΔABC in which AB = BC.
Prove that ΔDEF is also isosceles.
In triangle ABC; M is mid-point of AB, N is mid-point of AC and D is any point in base BC. Use the intercept Theorem to show that MN bisects AD.
If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral ABCD is a rectangle,
show that the diagonals AC and BD intersect at the right angle.
Prove that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.
In a parallelogram ABCD, E and F are the midpoints of the sides AB and CD respectively. The line segments AF and BF meet the line segments DE and CE at points G and H respectively Prove that: EGFH is a parallelogram.
In the given figure, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:
ST = `(1)/(3)"LS"`
D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC. Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent triangles.
