Advertisements
Advertisements
प्रश्न
In the Figure, `square`ABCD is a trapezium. AB || DC. Points P and Q are midpoints of seg AD and seg BC respectively. Then prove that, PQ || AB and PQ = `1/2 ("AB" + "DC")`.
Advertisements
उत्तर
Given: `square`ABCD is a trapezium.
To prove: PQ || AB and PQ = `1/2`(AB + DC)
Construction: Extend line AQ in such a way that, on extending side DC, intersect it at point R.
Proof:
seg AB || seg DC ...(Given)
and seg BC is their transversal.
∴ ∠ABC ≅ ∠RCB ...(Alternate angles)
∴ ∠ABQ ≅ ∠RCQ ...(i) ...(B-Q-C)
In ∆ABQ and ∆RCQ,
∠ABQ ≅∠RCQ ...[From (i)]
seg BQ ≅ seg CQ ...(Q is the midpoint of seg BC)
∠BQA ≅ ∠CQR ...(Vertically opposite angles)
∴ ∆ABQ ≅ ∆RCQ ...(ASA test)
seg AB ≅ seg CR ...(c.s.c.t.) ...(ii)
seg AQ ≅ seg RQ ...(c.s.c.t.) ...(iii)
In ∆ADR,
Point P is the midpoint of line AD. ...(Given)
Point Q is the midpoint of line AR. ...[From (iii)]
∴ seg PQ || side DR ...(Midpoint Theorem)
∴ seg PQ || side DC ...(iv) ...(D-C-R)
∴ side AB || side DC ...(v) ...(Given)
∴ seg PQ || side AB ...[From (iv) and (v)]
PQ = `1/2` DR ...(Midpoint Theorem)
= `1/2` (DC + CR)
= `1/2` (DC + AB) ...[From (ii)]
∴ PQ = `1/2` (AB + DC)
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 square E, F, G and H are points on AB, BC, CD and DA respectively, such that AE = BF = CG = DH. Prove that EFGH is a square.
In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC =
21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ.
Let Abc Be an Isosceles Triangle in Which Ab = Ac. If D, E, F Be the Mid-points of the Sides Bc, Ca and a B Respectively, Show that the Segment Ad and Ef Bisect Each Other at Right Angles.
ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH.
Show that the line segments joining the mid-points of the opposite sides of a quadrilateral
bisect each other.
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.
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.
The figure, given below, shows a trapezium ABCD. M and N are the mid-point of the non-parallel sides AD and BC respectively. Find:
- MN, if AB = 11 cm and DC = 8 cm.
- AB, if DC = 20 cm and MN = 27 cm.
- DC, if MN = 15 cm and AB = 23 cm.
In triangle ABC, AD is the median and DE, drawn parallel to side BA, meets AC at point E.
Show that BE is also a median.
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 is the mid-point of AB and E is the mid-point of BC.
Calculate:
(i) DE, if AC = 8.6 cm
(ii) ∠DEB, if ∠ACB = 72°
In ΔABC, D, E, F are the midpoints of BC, CA and AB respectively. Find ∠FDB if ∠ACB = 115°.
In a right-angled triangle ABC. ∠ABC = 90° and D is the midpoint of AC. Prove that BD = `(1)/(2)"AC"`.
Side AC of a ABC is produced to point E so that CE = `(1)/(2)"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 meets AC at point P and EF at point R respectively. Prove that: 3DF = EF
In ΔABC, X is the mid-point of AB, and Y is the mid-point of AC. BY and CX are produced and meet the straight line through A parallel to BC at P and Q respectively. Prove AP = AQ.
In ΔABC, the medians BE and CD are produced to the points P and Q respectively such that BE = EP and CD = DQ. Prove that: Q A and P are collinear.
The figure obtained by joining the mid-points of the sides of a rhombus, taken in order, is ______.
D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle ABC. Show that ∆DEF is also an equilateral triangle.
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.
