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In the given figure ABCD is a rectangle and the side lengths are integ 08 Jan 2022, 12:29
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75% (hard)

Question Stats:

56% (03:17) correct 44% (03:09) wrong based on 71 sessions

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In the given figure ABCD is a rectangle and the side lengths are integers. The perimeter of the rectangle is 48 and AB is parallel to the x-axis. If AB – BC < 4, which of the following can be the co-ordinate of C?

I. (16, 17)

II. (16, 15)

III. (17,16)

A. I only

B. II only

C. III only

D. I and III

E. I , II, and III
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D
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In the given figure ABCD is a rectangle and the side lengths are integ 18 Jan 2022, 23:00
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In the given figure ABCD is a rectangle and the side lengths are integers. The perimeter of the rectangle is 48 and AB is parallel to the x-axis. If AB – BC < 4, which of the following can be the co-ordinate of C?

I. (16, 17)

II. (16, 15)

III. (17,16)

A. I only

B. II only

C. III only

D. I and III

E. I , II, and III[/quote]


2*(AB+BC) =48..So, (AB+BC)=24...i.e., BC=24-AB..eqn 1
Given, AB-BC < 4
substituting eqn 1...
AB-(24-AB) < 4
AB<14.

Now, as sides are integers(given), max value of AB =13. When AB=13, BC =24-13 =11. Then, C = (4+13,5+11) =(17,16)..No. III
When AB=12, BC =24-12 =12. Then, C = (4+12,5+12) =(16,17)..No. I...
Now, if I is correct, II cannot be.

So, I & III is possible.
Option D
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In the given figure ABCD is a rectangle and the side lengths are integ 04 Jul 2022, 03:29
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if we consider the choices individually

I. (16, 17)

this means that length of \(AB = \sqrt{ (16-4)^2- (5-5)^2} = 12\)
ans length of BC = Y Cordinate of C - Y cordinate of \(B = 17-5=12 \)
These values of AB and BC satisfy the AB – BC < 4 and perimeter=48

II. (16, 15)
this means that length of \(AB = \sqrt{ (16-4)^2- (5-5)^2} = 12\)
ans length of BC = Y Cordinate of C - Y cordinate of \(B = 15-5=10\)
These values of AB and BC do not satisfy perimeter=48

III. (17,16)
this means that length of \(AB = \sqrt{ (17-4)^2- (5-5)^2} = 13\)
ans length of BC = Y Cordinate of C - Y cordinate of \(B = 16-5=11\)
These values of AB and BC satisfy the AB – BC < 4 and perimeter=48

only I and III satisfy both the conditions.
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In the given figure ABCD is a rectangle and the side lengths are integ 04 Jul 2022, 08:47
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AbhiroopGhosh
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In the given figure ABCD is a rectangle and the side lengths are integers. The perimeter of the rectangle is 48 and AB is parallel to the x-axis. If AB – BC < 4, which of the following can be the co-ordinate of C?

I. (16, 17)

II. (16, 15)

III. (17,16)

A. I only

B. II only

C. III only

D. I and III

E. I , II, and III


I. We go over 12 and up 12. That gives us a 12x12 square. Perimeter = 48 and AB-BC < 4. I works. B and C are out.

II. We go over 12 and up 10. That gives us a 12x10 rectangle. Perimeter does not equal 48. Eliminate II. D and E are out.

Answer choice A.

Just for completeness even though we already have the correct answer, III. We go over 13 and up 11. That gives us a 13x11 rectangle. Perimeter = 48 and AB-BC < 4. III works.
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In the given figure ABCD is a rectangle and the side lengths are integ 04 Mar 2025, 23:49
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In the given figure ABCD is a rectangle and the side lengths are integers. The perimeter of the rectangle is 48 and AB is parallel to the x-axis.

If AB – BC < 4, which of the following can be the co-ordinate of C?

Perimeter of the rectangle = 2(x-4) + 2(y-5) = 2x + 2y - 18 = 48;
x + y = 33

AB=x-4
BC=y-5

AB - BC < 4
(x-4) - (y-5) < 4
x - y < 3

2x < 36
x < 18

x = 33-y < 18
y > 33-18 = 15
y>15

I. (16, 17)

II. (16, 15)

III. (17,16)

x< 18 in all cases but y>15 is correct in I & III only

A. I only

B. II only

C. III only

D. I and III

E. I , II, and III

IMO D
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