In triangle ABC, the measure of angle B is 90°, the length of side AB

Question

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 ?

I. 1
II. 5
III. 6

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

I. 1
II. 5
III. 6

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III

Archit3110 · Answer

value of x will fall under range ; 64-16 ; √48 ; 4√3 ; 6.92
so 0-6.92= x
IMO E

nick1816 · Answer

4+x>\sqrt{(16+x^2)} x>0 Also 16<16+x^2<64 0

ScottTargetTestPrep · Answer

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.

Answer: E

Mck2023 · Answer

HI  Bunuel , nice question!!

Can you please help me clear my doubts.

The value of AC is between 4 and 8. In this case, should I assume the value of 4 and 8 is inclusive? Further, If not inclusive, can I assume the value of AC be 4.5 (an non integer) or the value has to be an integer? Thank You!!!

Bunuel · Answer

1. Between 4 and 8 means not inclusive.
2. Yes, the length of AC can be any number between 4 and 8, not necessarily an integer.

Ahasunhabib999 · Answer

Hi, 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.

paU1i · Answer

I was thinking the same thing. Sum of any two sides should always be greater than third side and difference should always be less than third side. By this logic only II and III fit in

Am I thinking something wrong?

Bunuel

In triangle ABC, the measure of angle B is 90°, the length of side AB

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In triangle ABC, the measure of angle B is 90°, the length of side AB

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