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प्रश्न
Four alternative answers for the following question is given. Choose the correct alternative.
A circle touches all sides of a parallelogram. So the parallelogram must be a, ______
पर्याय
rectangle
rhombus
square
trapezium
उत्तर
ABCD is a parallelogram. A circle with centre O touches the parallelogram at E, F, G and H.
ABCD is a parallelogram.
∴ AB = CD .....(1) (Opposite sides of parallelogram are equal)
AD = BC .....(2) (Opposite sides of parallelogram are equal)
Tangent segments drawn from an external point to a circle are congruent.
AE = AH .....(3)
DG = DH .....(4)
BE = BF .....(5)
CG = CF .....(6)
Adding (3), (4), (5) and (6), we get
AE + BE + CG + DG = AH + DH + BF + CF
⇒ AB + CD = AD + BC .....(7)
From (1), (2) and (7), we ahve
2AB = 2BC
⇒ AB = BC .....(8)
From (1), (2) and (8), we have
AB = BC = CD = AD
∴ Parallelogram ABCD is a rhombus. (A rhombus is a parallelogram with all sides equal)
Hence, the correct answer is rhombus .
संबंधित प्रश्न
If two circles with radii 8 cm and 3 cm, respectively, touch internally, then find the distance between their centers.
If two circles with radii 8 cm and 3 cm respectively touch externally, then find the distance between their centres.
Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres.
Four alternative answers for the following question is given. Choose the correct alternative.
Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres
If radii of two circles are 4 cm and 2.8 cm. Draw a figure of this circles touching each other externally.
In the given figure, the circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that – (1) seg AP || seg BQ,
(2) ∆APR ~ ∆RQB, and
(3) Find ∠ RQB if ∠ PAR = 35°
In the given figure, ray PQ touches the circle at point Q. PQ = 12, PR = 8, find PS and RS.
Four alternative answers for the following question is given. Choose the correct alternative.
Chords AB and CD of a circle intersect inside the circle at point E. If AE = 5.6, EB = 10, CE = 8, find ED.
Line l touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following.
- What is d(O, P) = ? Why?
- If d(O, Q) = 8 cm, where does the point Q lie?
- If d(O, Q) = 15 cm, How many locations of point Q are line on line l? At what distance will each of them be from point P?
In the given figure, circle with centre M touches the circle with centre N at point T. Radius RM touches the smaller circle at S. Radii of circles are 9 cm and 2.5 cm. Find the answers to the following questions hence find the ratio MS:SR.
(1) Find the length of segment MT
(2) Find the length of seg MN
(3) Find the measure of ∠NSM.
In the adjoining figure circles with centres X and Y touch each other at point Z. A secant passing through Z intersects the circles at points A and B respectively. Prove that, radius XA || radius YB. Fill in the blanks and complete the proof.
Construction: Draw segments XZ and YZ.
Proof:
By theorem of touching circles, points X, Z, Y are `square`.
∴ ∠XZA ≅ `square` ...(opposite angles)
Let ∠XZA = ∠BZY = a ...(I)
Now, seg XA ≅ seg XZ ...[Radii of the same circle]
∴∠XAZ = `square` = a ...[isosceles triangle theorem](II)
Similarly,
seg YB ≅ seg YZ ...[Radii of the same circle]
∴∠BZY = `square` = a ...[isosceles triangle theorem](III)
∴ from (I), (II), (III),
∠XAZ = `square`
∴ radius XA || radius YZ ...[`square`]
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Line ℓ touches a circle with centre O at point P. If radius of the circle is 9 cm, answer the following.
What is d(O, P) =? Why?
Line ℓ touches a circle with center O at point P. If the radius of the circle is 9 cm, answer the following.
If d(O, Q) = 8 cm, where does the point Q lie?
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