Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
In a ∆ABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are 7 cm, 8 cm and 9 cm, respectively, find the perimeter of ∆DEF.
BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If
L is the mid-point of BC, prove that LM = LN.
Fill in the blank to make the following statement correct
The triangle formed by joining the mid-points of the sides of an isosceles triangle is
In the given figure, seg PD is a median of ΔPQR. Point T is the mid point of seg PD. Produced QT intersects PR at M. Show that `"PM"/"PR" = 1/3`.
[Hint: DN || QM]
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.
In a triangle ABC, AD is a median and E is mid-point of median AD. A line through B and E meets AC at point F.
Prove that: AC = 3AF.
In trapezium ABCD, AB is parallel to DC; P and Q are the mid-points of AD and BC respectively. BP produced meets CD produced at point E.
Prove that:
- Point P bisects BE,
- PQ is parallel to AB.
In triangle ABC, angle B is obtuse. D and E are mid-points of sides AB and BC respectively and F is a point on side AC such that EF is parallel to AB. Show that BEFD is a parallelogram.
In ΔABC, D, E, F are the midpoints of BC, CA and AB respectively. Find DE, if AB = 8 cm
Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
In a right-angled triangle ABC. ∠ABC = 90° and D is the midpoint of AC. Prove that BD = `(1)/(2)"AC"`.
In the given figure, ABCD is a trapezium. P and Q are the midpoints of non-parallel side AD and BC respectively. Find: AB, if DC = 8 cm and PQ = 9.5 cm
ABCD is a kite in which BC = CD, AB = AD. E, F and G are the mid-points of CD, BC and AB respectively. Prove that: ∠EFG = 90°
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.
In AABC, D and E are two points on the side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet the side AC at points F and G respectively. Through F and G, lines are drawn parallel to AB which meet the side BC at points M and N respectively. Prove that BM = MN = NC.
In the given figure, PS = 3RS. M is the midpoint of QR. If TR || MN || QP, then prove that:
RT = `(1)/(3)"PQ"`
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if ______.
E is the mid-point of the side AD of the trapezium ABCD with AB || DC. A line through E drawn parallel to AB intersect BC at F. Show that F is the mid-point of BC. [Hint: Join AC]
P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus.
