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In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
Updated on: 12 Jul 2021, 05:43

2

Expert Reply

5

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Question Stats:

In triangle ABC, the measure of angle B is 90°, the length of side AB is 4, and the length of side BC is x. If the length of hypotenuse AC is between 4 and 8, which of the following could be the value of x ?

Indicate \(all\) such values.

❑ 1

❑ 2

❑ 3

❑ 4

❑ 5

❑ 6

Indicate \(all\) such values.

❑ 1

❑ 2

❑ 3

❑ 4

❑ 5

❑ 6

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Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
27 Nov 2017, 16:46

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1

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Look at the figure below:

Length of side AB is 4, and the length of side BC is xand AC is between 4 and 8. There can be many triangles that satify this condition.

On one extreme when AC is almost 4. Then the the triangle becomes almost the line AB and \(x\) is almost 0.

On the other hand when AC is 8 then \(x=\sqrt{8^2-4^2} \approx 6.9\).

So all values from 0 to 6.9 form a right triangle. Hence all options are correct!

right.jpg [ 13.28 KiB | Viewed 26623 times ]

_________________

Length of side AB is 4, and the length of side BC is xand AC is between 4 and 8. There can be many triangles that satify this condition.

On one extreme when AC is almost 4. Then the the triangle becomes almost the line AB and \(x\) is almost 0.

On the other hand when AC is 8 then \(x=\sqrt{8^2-4^2} \approx 6.9\).

So all values from 0 to 6.9 form a right triangle. Hence all options are correct!

Show: :: img

Attachment:

right.jpg [ 13.28 KiB | Viewed 26623 times ]

_________________

Sandy

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Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
12 Jul 2021, 05:40

2

Expert Reply

Carcass wrote:

In triangle ABC, the measure of angle B is 90°, the length of side AB is 4, and the length of side BC is x. If the length of hypotenuse AC is between 4 and 8, which of the following could be the value of x ?

Indicate \(all\) such values.

❑ 1

❑ 2

❑ 3

❑ 4

❑ 5

❑ 6

Indicate \(all\) such values.

❑ 1

❑ 2

❑ 3

❑ 4

❑ 5

❑ 6

I would solve the question by testing each choice.

A) x = 1

The Pythagorean theorem tells us that 4² + 1² = hypotenuse²

Solve to get: Hypotenuse = √17

Since √17 is between 4 and 8, x COULD equal 1

B) x = 2

The Pythagorean theorem tells us that 4² + 2² = hypotenuse²

Solve to get: Hypotenuse = √20

Since √20 is between 4 and 8, x COULD equal 2

C) x = 3

The Pythagorean theorem tells us that 4² + 3² = hypotenuse²

Solve to get: Hypotenuse = √25

Since √25 is between 4 and 8, x COULD equal 3

D) x = 4

The Pythagorean theorem tells us that 4² + 4² = hypotenuse²

Solve to get: Hypotenuse = √32

Since √32 is between 4 and 8, x COULD equal 4

E) x = 5

The Pythagorean theorem tells us that 4² + 5² = hypotenuse²

Solve to get: Hypotenuse = √41

Since √41 is between 4 and 8, x COULD equal 5

F) x = 6

The Pythagorean theorem tells us that 4² + 6² = hypotenuse²

Solve to get: Hypotenuse = √52

Since √52 is between 4 and 8, x COULD equal 6

Answer: A, B, C, D, E and F

_________________

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
03 Mar 2021, 16:24

1

I always get confused when it says "between" does it mean that the hypotenuse is greater than 4 and less than 8? Or does it mean that the hypotenuse is greater than or equal to four and less than or equal to 8?

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
14 May 2021, 08:16

Hi sandy, but should we not consider the property "the sum of two sides should be greater than the third side" while selecting the options?

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
14 May 2021, 10:50

1

Expert Reply

Fixed the OAs which were wrong for some reason

By the Pythagorean theorem AB^2 + BC^2 = AC^2 or 4^2 + x^2 = AC^2. If AC = 4, then we have:

4^2 + x^2 = 4^2

x^2 = 0

x = 0

Of course, x can’t be 0. However, recall that AC is actually greater than 4, so x is actually greater than 0.

If AC = 8, then we have:

4^2 + x^2 = 8^2

16 + x^2 = 64

x^2 = 48

x = √48 = 4√3

Recall that AC is actually less than 8, so x is actually less than 4√3.

Since 0 < x < 4√3, x can be 1, 5 or 6.

Answers are : A, E, and F

By the Pythagorean theorem AB^2 + BC^2 = AC^2 or 4^2 + x^2 = AC^2. If AC = 4, then we have:

4^2 + x^2 = 4^2

x^2 = 0

x = 0

Of course, x can’t be 0. However, recall that AC is actually greater than 4, so x is actually greater than 0.

If AC = 8, then we have:

4^2 + x^2 = 8^2

16 + x^2 = 64

x^2 = 48

x = √48 = 4√3

Recall that AC is actually less than 8, so x is actually less than 4√3.

Since 0 < x < 4√3, x can be 1, 5 or 6.

Answers are : A, E, and F

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
15 May 2021, 22:46

1

Carcass wrote:

Fixed the OAs which were wrong for some reason

By the Pythagorean theorem AB^2 + BC^2 = AC^2 or 4^2 + x^2 = AC^2. If AC = 4, then we have:

4^2 + x^2 = 4^2

x^2 = 0

x = 0

Of course, x can’t be 0. However, recall that AC is actually greater than 4, so x is actually greater than 0.

If AC = 8, then we have:

4^2 + x^2 = 8^2

16 + x^2 = 64

x^2 = 48

x = √48 = 4√3

Recall that AC is actually less than 8, so x is actually less than 4√3.

Since 0 < x < 4√3, x can be 1, 5 or 6.

Answers are : A, E, and F

By the Pythagorean theorem AB^2 + BC^2 = AC^2 or 4^2 + x^2 = AC^2. If AC = 4, then we have:

4^2 + x^2 = 4^2

x^2 = 0

x = 0

Of course, x can’t be 0. However, recall that AC is actually greater than 4, so x is actually greater than 0.

If AC = 8, then we have:

4^2 + x^2 = 8^2

16 + x^2 = 64

x^2 = 48

x = √48 = 4√3

Recall that AC is actually less than 8, so x is actually less than 4√3.

Since 0 < x < 4√3, x can be 1, 5 or 6.

Answers are : A, E, and F

what about option b,c,d?

Since 0 < x < 4√3,

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
08 Jul 2021, 21:34

1

Did we find out the correct answers here? Is it all the options or A,E,F?

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
09 Jul 2021, 01:38

1

Expert Reply

I am not quite sure what you meant

However, there is the timer and at the bottom the spoiler: the correct answers are A,E,F

However, there is the timer and at the bottom the spoiler: the correct answers are A,E,F

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
Updated on: 11 Jul 2021, 10:35

Hi Carcass, should we not consider the property "the sum of two sides should be greater than the third side" while selecting the options?

If we consider the two sides of the triangle tobe 4 and 1 , by Pythagorus theorem we get the hypotenuse to be 4.123 . My question is if we sum 2 sides shouldn’t it be always greater than third side. Please clarify. So for 2 it is also justified here if we calculate like this.

_________________

If we consider the two sides of the triangle tobe 4 and 1 , by Pythagorus theorem we get the hypotenuse to be 4.123 . My question is if we sum 2 sides shouldn’t it be always greater than third side. Please clarify. So for 2 it is also justified here if we calculate like this.

_________________

Life is 10% what happens to you and 90% how you react to it. ...

#GRE_Student

#GRE_Student

Originally posted by Ahasunhabib999 on 11 Jul 2021, 09:55.

Last edited by Ahasunhabib999 on 11 Jul 2021, 10:35, edited 1 time in total.

Last edited by Ahasunhabib999 on 11 Jul 2021, 10:35, edited 1 time in total.

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
11 Jul 2021, 10:01

Expert Reply

Ahasunhabib999 wrote:

Hi Carcass, should we not consider the property "the sum of two sides should be greater than the third side" while selecting the options?

If we consider the two sides of the triangle tobe 4 and 1 , by Pythagorus theorem we get the hypotenuse to be 4.123 . My question is if we sum 2 sides shouldn’t it be greater than third side. Please clarify.

If we consider the two sides of the triangle tobe 4 and 1 , by Pythagorus theorem we get the hypotenuse to be 4.123 . My question is if we sum 2 sides shouldn’t it be greater than third side. Please clarify.

I DO NOT THINK IN THIS CASE, FRANKLY.

uSUALLY, WE USE THAT STATEMENT OR TRUTH WHEN WE do have a fixed value and we do know the two values

Here we do have a range, so in my view is a waste of time using that.

You could go down the wrong path

Re: In triangle ABC, the measure of angle B is 90°, the length o
[#permalink]
11 Jul 2021, 10:27

1

Dear Carcass, Why option BCD isnot the valid answers here. Sorry i am unable to understand it.

_________________

_________________

Life is 10% what happens to you and 90% how you react to it. ...

#GRE_Student

#GRE_Student

gmatclubot

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