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
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.
ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that
- D is the mid-point of AC
- MD ⊥ AC
- CM = MA = `1/2AB`
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 ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing
through A. If L is the mid-point of BC, prove that ML = NL.
In Fig. below, BE ⊥ AC. AD is any line from A to BC intersecting BE in H. P, Q and R are
respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°.
ABCD is a kite having AB = AD and BC = CD. Prove that the figure formed by joining the
mid-points of the sides, in order, is a rectangle.
In the given figure, points X, Y, Z are the midpoints of side AB, side BC and side AC of ΔABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
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 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.
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, BE and CF are medians. P is a point on BE produced such that BE = EP and Q is a point on CF produced such that CF = FQ. Prove that: QAP is a straight line.
AD is a median of side BC of ABC. E is the midpoint of AD. BE is joined and produced to meet AC at F. Prove that AF: AC = 1 : 3.
In ΔABC, D, E and F are the midpoints of AB, BC and AC.
Show that AE and DF bisect each other.
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: A is the mid-point of PQ.
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, T is the midpoint of QR. Side PR of ΔPQR is extended to S such that R divides PS in the ratio 2:1. TV and WR are drawn parallel to PQ. Prove that T divides SU in the ratio 2:1 and WR = `(1)/(4)"PQ"`.
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if ______.
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.
P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC ⊥ BD. Prove that PQRS is a square.
E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and EF = `1/2` (AB + CD).
[Hint: Join BE and produce it to meet CD produced at G.]
