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The sides of a triangle are 1, x, and x^2. What are possi
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30 Nov 2018, 17:14

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Question Stats:

The sides of a triangle are 1, x, and \(x^2\). Whthe at are possible values of x?

Indicate all possible values.

A. 0.5

B. 1

C. 1.5

D. 2

E. 2.5

F. 3

G. 3.5

Indicate all possible values.

A. 0.5

B. 1

C. 1.5

D. 2

E. 2.5

F. 3

G. 3.5

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Re: The sides of a triangle are 1, x, and x^2. What are possi
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10 Dec 2018, 10:54

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Carcass wrote:

The sides of a triangle are 1, x, and \(x^2\). What are possible values of x? Indicate all possible values.

A. 0.5

B. 1

C. 1.5

D. 2

E. 2.5

F. 3

G. 3.5

A. 0.5

B. 1

C. 1.5

D. 2

E. 2.5

F. 3

G. 3.5

IMPORTANT RULE: If two sides of a triangle have lengths A and B, then: |A-B| < third side < A+B

Now let's see what measurements we get for each value of x.

A. x = 0.5. So 1, x, and x² become 1, 0.5 and 0.25. Here, it is NOT the case that |0.5 - 0.25| < 1 < 0.5 + 0.25

ELIMINATE A.

B. x = 1. So 1, x, and x² become 1, 1 and 1. Here, it IS the case that |1 - 1| < 1 < 1 + 1

B WORKS

C. x = 1.5. So 1, x, and x² become 1, 1.5 and 2.25. Here, it IS the case that |1.5 - 1| < 2.25 < 1 + 1.5

C WORKS

D. x = 2. So 1, x, and x² become 1, 2 and 4. Here, it is NOT the case that |1 - 2| < 4 < 1 + 2

ELIMINATE D.

E. x = 2.5. So 1, x, and x² become 1, 2.5 and 6.25

Here, it is NOT the case that |1 - 2.5| < 6.25 < 1 + 2.5

ELIMINATE E.

F. x = 3. So 1, x, and x² become 1, 3 and 9. Here, it is NOT the case that |1 - 3| < 9 < 1 + 3

ELIMINATE F.

G. x = 3.5. So 1, x, and x² become 1, 3.5 and 12.25. Here, it is NOT the case that |1 - 3.5| < 12.25 < 1 + 3.5

ELIMINATE G.

Answer: B, C

Cheers,

Brent

The sides of a triangle are 1, x, and x2. What are the possi
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14 Mar 2019, 09:47

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Expert Reply

1

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The sides of a triangle are \(1\), \(x\), and \(x^2\). What are the possible values of x?

Indicate all possible values.

A. 0.5

B. 1

C. 1.5

D. 2

E. 2.5

F. 3

G. 3.5

Indicate all possible values.

A. 0.5

B. 1

C. 1.5

D. 2

E. 2.5

F. 3

G. 3.5

Re: The sides of a triangle are 1, x, and x2. What are the possi
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20 Mar 2019, 16:25

1

Expert Reply

This one will test the triangle inequality theorem, which states that the sum of any two sides of a triangle must be larger than the third side.

If x is less than 1, then 1 is the largest side. So x + x² must be greater than 1. When x = 0.5, though, x² = 0.25, and 0.5 + 0.25 is not greater than 1. So x = 0.5 doesn't work.

1 is fine, as it creates an equilateral triangle.

In anything greater than 1, x² will be the longest side. So we must satisfy the inequality x² < 1 + x.

When x = 1.5, everything barely holds up:

(1.5)² < 1 + 1.5

2.25 < 2.5

At x = 2, though, x² becomes greater than 1 + x. So anything 2 or greater will not work. Only B and C create possible triangles.

If x is less than 1, then 1 is the largest side. So x + x² must be greater than 1. When x = 0.5, though, x² = 0.25, and 0.5 + 0.25 is not greater than 1. So x = 0.5 doesn't work.

1 is fine, as it creates an equilateral triangle.

In anything greater than 1, x² will be the longest side. So we must satisfy the inequality x² < 1 + x.

When x = 1.5, everything barely holds up:

(1.5)² < 1 + 1.5

2.25 < 2.5

At x = 2, though, x² becomes greater than 1 + x. So anything 2 or greater will not work. Only B and C create possible triangles.

Re: The sides of a triangle are 1, x, and x2. What are the possi
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06 Jan 2022, 09:40

Expert Reply

Re: The sides of a triangle are 1, x, and x^2. What are possi
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22 Jul 2024, 12:07

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Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

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Re: The sides of a triangle are 1, x, and x^2. What are possi [#permalink]

22 Jul 2024, 12:07
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