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The triangle shown above is Obtuse, where AC is the longest side and A
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30 Apr 2023, 23:10

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The triangle shown above is Obtuse, where AC is the longest side and AB is the shortest side .What are all possible lengths of AB if k is a positive integer?

Indicate all that apply

3

4

5

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Re: The triangle shown above is Obtuse, where AC is the longest side and A
[#permalink]
28 May 2023, 00:44

Expert Reply

OE

Given that, ABC is an obtuse angled triangle, where AC is the longest side and AB is the

shortest side.

In a triangle,

(Difference of other two sides) < Any side of the triangle < (Sum of other two sides)

So,

3k – k < 10 < 3k + k

2k < 10 < 4k

Therefore, 2k < 10 and 10 < 4k

that is, k < 5 and 2.5 < k

Therefore, k lies between 2.5 and 5.

It is given that k is an integer,

Therefore, K can be 3 or 4.

If k = 3,

AC = 3k = 9

But it is given that AC is longest side

Hence, k cannot be 3.

If k = 4,

AB = 4 and AC = 3k = 12

This satisfies the statements given in the question.

The only value of k possible is 4.

Ans. (B)

_________________

Given that, ABC is an obtuse angled triangle, where AC is the longest side and AB is the

shortest side.

In a triangle,

(Difference of other two sides) < Any side of the triangle < (Sum of other two sides)

So,

3k – k < 10 < 3k + k

2k < 10 < 4k

Therefore, 2k < 10 and 10 < 4k

that is, k < 5 and 2.5 < k

Therefore, k lies between 2.5 and 5.

It is given that k is an integer,

Therefore, K can be 3 or 4.

If k = 3,

AC = 3k = 9

But it is given that AC is longest side

Hence, k cannot be 3.

If k = 4,

AB = 4 and AC = 3k = 12

This satisfies the statements given in the question.

The only value of k possible is 4.

Ans. (B)

_________________

Re: The triangle shown above is Obtuse, where AC is the longest side and A
[#permalink]
13 Jul 2023, 01:57

2

Pick the the number 4 what only satisfies the conditions about the triangle's sides- AC as the longest side and any two sides sum and difference are more and less respectively than the other side.

Re: The triangle shown above is Obtuse, where AC is the longest side and A
[#permalink]
16 Jul 2023, 02:20

1

Difference between 2 sides < third side < Sum of 2 sides of a triangle

Side 1 = k

Side 2 = 3k

Side 3 = 10

(3k-k) < 10 < (3k+k)

2k < 10 < 4k

5 < 10 < 2.5

Since the required value is an integer, the only possible solution is 4.

Side 1 = k

Side 2 = 3k

Side 3 = 10

(3k-k) < 10 < (3k+k)

2k < 10 < 4k

5 < 10 < 2.5

Since the required value is an integer, the only possible solution is 4.

gmatclubot

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