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The two sides of a right triangle are 12 and 15. Which one of the foll

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Bunuel
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The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   02 Jul 2020, 02:39
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The two sides of a right triangle are 12 and 15. Which one of the following can be the length of the third side?

I. 9

II. \(3\sqrt{41}\)

III. 20

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III
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C
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Re: The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   02 Jul 2020, 02:46
Hypotenuse is the longest side of a right angled triangle.

So, there are two triangles possible.
Case 1

Base = 12 and Perpendicular = 15
Third side = Hypotenuse \( = \sqrt{15^2 + 12^2} = \sqrt{225 + 144} = \sqrt{369} \)

Hypotenuse = \(\sqrt{3^2 \times 41} = 3 \times \sqrt{41}\)

Case 2

Hypotenuse = 15 and one of the Base or perpendicular = 12.

Third Side \( = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9\)

OA, C
Bunuel wrote:
The two sides of a right triangle are 12 and 15. Which one of the following can be the length of the third side?

I. 9

II. \(3\sqrt{41}\)

III. 20

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III
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Re: The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   02 Jul 2020, 03:01
Explanation:

Let H=15, B or P=12
So, H^2 = B^2 + P^2
225 = 144 + P^2
P^2 = 81
P = 9

Let B=12, P=15 ,H=?
So, H^2 = B^2 + P^2
H^2 = 144 + 225
H^2 = 369
H = 3√41
Option 1 & 2 Satisfies

IMO-C
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Re: The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   05 Jul 2020, 06:48
Bunuel wrote:
The two sides of a right triangle are 12 and 15. Which one of the following can be the length of the third side?

I. 9

II. \(3\sqrt{41}\)

III. 20

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III



Given sides: 12 , 15
Given Triangle: Right angled.

Possible Combinations:

1. 12, 15 are given we calculate the hypotenuse.
=> \(\sqrt{12^2 + 15^2}\) = \(\sqrt{369}\) = \(3\sqrt{41}\)
=> II could be one of the sides.

2. we let 15 be the hypotenuse, and calculate the other side.
=> \(\sqrt{15^2 - 12^2}\) = \(\sqrt{81}\) = 9.
=> I could also be a side.

But III can never be a side, since the largest side of the triangle could either be 15 or \(3\sqrt{41}\).

Hence, Answer is C.
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The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   05 Jul 2020, 08:13
Bunuel wrote:
The two sides of a right triangle are 12 and 15. Which one of the following can be the length of the third side?

I. 9

II. \(3\sqrt{41}\)

III. 20

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III


The simplest way to solve is the use of Pythagoras theorem

As the 2 sides given 12 and 15 the other side either be smaller than 15 or greater than 15.

if it is less than 15, it acts as either height or base of the triangle and if it is greater than 15, it acts as the hypotenuse of the triangle.

I. 9 less than 15

\(\sqrt{9^2+12^2}\) check if you are getting 15 (YES)

II. \(3\sqrt{41}\) =18 approx which is greater than 15

\(\sqrt{15^2+12^2}\) check if you are getting \(3\sqrt{41}\) (YES)

III. 20 is also greater than 15 which obviously gonna be incorrect because we already found out in II option, \(\sqrt{15^2+12^2}\) cannot be equal to 20

Hence answer C
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The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   05 Jul 2020, 09:05
Expert Reply
Bunuel wrote:
The two sides of a right triangle are 12 and 15. Which one of the following can be the length of the third side?

I. 9

II. \(3\sqrt{41}\)

III. 20

(A) I only
(B) III only
(C) I and II only
(D) I and III only
(E) I, II, and III



Property: Sum of any two sides of ∆ > Third Side of the ∆


Two sides are 12 and 15

If third side (x) is longest then, 12+15 > x i.e. x < 27

If third side (x) is NOT longets then, 12+x > 15 i.e. x > 3

i.e. 3 < x < 27

Point to note: Triangle given here is a right triangle

Case 1: \(12^2 + 15^2 = x^2\) i.e \(x = 3√41\)

Case 2: \(12^2 + x^2 = 15^2\) i.e \(x = 9\)


I. 9 POSSIBLE

II. \(3\sqrt{41}\) POSSIBLE

III. 20 NOT POSSIBLE

Answer: Option C
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The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink] New post   05 Jul 2020, 09:24
By pythogoras theorem
Sum of squares of perpendicular and base is equal to hypotenuse square.

Case one when both 12 and 15 are perpendicular and base then the third side hypotenuse will be option 2. And in other case when 15 I'd hypotenuse then the three sides will be 12,9 and 15.
So both 1 and 2 are correct

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The two sides of a right triangle are 12 and 15. Which one of the foll [#permalink]
05 Jul 2020, 09:24
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