A circle touches all sides of a parallelogram. So the parallelogram must be a, ______ rectangle rhombus square trapezium
Chapter 3 – Circle – Text Book Solution
Problem Set 3 | Q 1.03 | Page 83
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, ______
- rectangle
- rhombus
- square
- trapezium
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 .
Explanation:-
In this problem, we have a parallelogram ABCD, and a circle with center O that touches the parallelogram at points E, F, G, and H. We need to determine the shape of the parallelogram.
We start by using the properties of parallelograms. From the definition of a parallelogram, we know that opposite sides are equal. Therefore, AB = CD and AD = BC (equations 1 and 2).
We also know that tangent segments drawn from an external point to a circle are congruent. From this property, we get that AE = AH, DG = DH, BE = BF, and CG = CF (equations 3-6).
Adding equations 3-6, we get:
AE + BE + CG + DG = AH + DH + BF + CF AB + CD = AD + BC ……(7)
Using equations 1, 2, and 7, we get:
2AB = 2BC AB = BC …….(8)
From equations 1, 2, and 8, we have:
AB = BC = CD = AD
Therefore, parallelogram ABCD has all sides equal, which means it is a rhombus.
Hence, the correct answer is a rhombus.
Chapter 3 – Circle – Text Book Solution
Problem Set 3 | Q 1.03 | Page 83
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