Advertisements
Advertisements
प्रश्न
In the given figure, G is the point of concurrence of medians of ΔDEF. Take point H on ray DG such that D-G-H and DG = GH, then prove that `square`GEHF is a parallelogram.
Advertisements
उत्तर
Let, the median side drawn from point D intersects EF at point M.
Point G is the concurrence point.
The concurrence point divides each median in the ratio 2:1.
∴ DG : GM = 2 : 1
∴ `("DG")/("GM") = 2/1`
∴ DG = 2GM ...(i)
∴ DG = GM + MH ...(G-M-H)
∴ 2GM = GM + MH ...[from (i)]
∴ 2GM – GM = MH
∴ GM = MH ...(ii)
In `square`GEHF,
Line GM ≅ Line MH ...[From (ii)]
Line EM ≅ Line MF ...(Point M is the midpoint of line EF)
A quadrilateral is a parallelogram if its diagonals bisect each other.
`square`GEHF is a parallelogram.
APPEARS IN
संबंधित प्रश्न
Consider the given parallelograms. Find the values of the unknowns x, y, z.
The following figure GUNS is a parallelogram. Find x and y. (Lengths are in cm)
Find the measure of ∠P and ∠S, if `bar(SP) || bar(RQ)` in the following figure. (If you find m∠R, is there more than one method to find m∠P?).
Construct ☐ PQRS, such that l(PQ) = 3.5 cm, l(QR) = 5.6 cm, l(RS) = 3.5 cm, m∠Q = 110°, m∠R = 70°. If it is given that ☐ PQRS is a parallelogram, which of the given information is unnecessary?
Construct a parallelogram ABCD such that l(BC) = 7 cm, m∠ABC = 40° , l(AB) = 3 cm.
In parallelogram ABCD, ∠A = 3 times ∠B. Find all the angles of the parallelogram. In the same parallelogram, if AB = 5x – 7 and CD = 3x +1 ; find the length of CD.
ABCD is a parallelogram. What kind of quadrilateral is it if : AC = BD and AC is perpendicular to BD?
In parallelogram ABCD, E is the mid-point of side AB and CE bisects angle BCD. Prove that:
- AE = AD,
- DE bisects and ∠ADC and
- Angle DEC is a right angle.
In parallelogram ABCD, X and Y are midpoints of opposite sides AB and DC respectively. Prove that:
(i) AX = YC
(ii) AX is parallel to YC
(iii) AXCY is a parallelogram.
The given figure shows parallelogram ABCD. Points M and N lie in diagonal BD such that DM = BN.
Prove that:
(i) ∆DMC = ∆BNA and so CM = AN
(ii) ∆AMD = ∆CNB and so AM CN
(iii) ANCM is a parallelogram.
Use the information given in the alongside diagram to find the value of x, y, and z.
If a triangle and a parallelogram lie on the same base and between the same parallels, then prove that the area of the triangle is equal to half of the area of parallelogram
Which of the following is a property of a parallelogram?
If two adjacent angles of a parallelogram are (5x – 5)° and (10x + 35)°, then the ratio of these angles is ______.
In parallelogram LOST, SN ⊥ OL and SM ⊥ LT. Find ∠STM, ∠SON and ∠NSM.
ABCD is a parallelogram. Find the value of x, y and z.
ABCD is a parallelogram. The bisector of angle A intersects CD at X and bisector of angle C intersects AB at Y. Is AXCY a parallelogram? Give reason.
The angle between the two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 45°. Find the angles of the parallelogram.
