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

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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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Official Answer

A,B,C,D,E,F

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

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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 36274 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 36274 times ]

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

3

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
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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

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Re: In triangle ABC, the measure of angle B is 90°, the length o
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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.

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
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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

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

1

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

Re: In triangle ABC, the measure of angle B is 90°, the length o
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22 Jun 2022, 02:16

1

GreenlightTestPrep wrote:

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

hello GreenlightTestPrep

i checked other posts, but sill could not find answer to my question.

why are not we considering "the sum of two sides should be greater than the third side" in case x = 1 and hypotenuse is between 4 and 8, then even 1 cant be answer because 4+1=5 which can be longer than hypotenuse say hypotenuse is 4.5 which is less than sum (4+1=5) of other two sides. and that violates the property of "the sum of two sides should be greater than the third side"

can you pls explain. thanks

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Re: In triangle ABC, the measure of angle B is 90°, the length o
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22 Jun 2022, 06:09

testtaker123 wrote:

hello GreenlightTestPrep

i checked other posts, but sill could not find answer to my question.

why are not we considering "the sum of two sides should be greater than the third side" in case x = 1 and hypotenuse is between 4 and 8, then even 1 cant be answer because 4+1=5 which can be longer than hypotenuse say hypotenuse is 4.5 which is less than sum (4+1=5) of other two sides. and that violates the property of "the sum of two sides should be greater than the third side"

can you pls explain. thanks

i checked other posts, but sill could not find answer to my question.

why are not we considering "the sum of two sides should be greater than the third side" in case x = 1 and hypotenuse is between 4 and 8, then even 1 cant be answer because 4+1=5 which can be longer than hypotenuse say hypotenuse is 4.5 which is less than sum (4+1=5) of other two sides. and that violates the property of "the sum of two sides should be greater than the third side"

can you pls explain. thanks

There is more than one rule at play here.

For example, since ABC is a right triangle, the Pythagorean theorem must hold true.

Also, since ABC is a triangle, "the sum of two sides should be greater than the third side" rule also must hold true.

If x = 1 (aka side BC), then: length of AC < 1 + 4. Since the length of AC is between 4 and 8, the rule holds true.

If x = 2 (aka side BC), then: length of AC < 2 + 4. Since the length of AC is between 4 and 8, the rule holds true.

If x = 3 (aka side BC), then: length of AC < 3 + 4. Since the length of AC is between 4 and 8, the rule holds true.

If x = 4 (aka side BC), then: length of AC < 4 + 4. Since the length of AC is between 4 and 8, the rule holds true.

If x = 5 (aka side BC), then: length of AC < 5 + 4. Since the length of AC is between 4 and 8, the rule holds true.

If x = 6 (aka side BC), then: length of AC < 6 + 4. Since the length of AC is between 4 and 8, the rule holds true.

Does that help?

Re: In triangle ABC, the measure of angle B is 90°, the length o
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26 Jun 2023, 12:36

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

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