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Is the triangle ABC with sides a, b,

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Is the triangle ABC with sides a, b,  [#permalink]

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New post Updated on: 14 Jul 2020, 08:30
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GMATBusters’ Quant Quiz Question -5


Is the triangle ABC with sides a, b, c an acute angles triangle?
1) The Sides AB (length = c) and BC (length = a) are 13 and 17 respectively and the angle ABC is 73.8 deg.
2) The triangle with sides a^2, b^2 and c^2 is a right-angled triangle.

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Originally posted by GMATBusters on 11 Jul 2020, 18:20.
Last edited by GMATBusters on 14 Jul 2020, 08:30, edited 2 times in total.
Revised angle to remove the contradiction between Statements, the solution/ answer has no effect.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 13 Jul 2020, 19:16
1
This Question is based on the logic that if you can draw a figure uniquely by using the information given, we can find the detail of that figure.
Image
Image
Image
Image

CONCEPT: How to find the nature of angles in a triangle.


If \(a^2+b^2 < c^2\), it means the angle between side a & b is obtuse, the triangle is obtuse.
If \(a^2+b^2 > c^2\), it means the angle between a & b is acute, but since other angle can be obtuse, the triangle cant be taken as acute.
If \(a^2+b^2 =c^2\), it means the angle between a & b is 90 deg, hence the triangle is right.

Must Do similar questions based on same logic:
1) https://gmatclub.com/forum/is-triangle-abc-with-sides-a-b-and-c-acute-angled-263435.html?hilit=acute
2) https://gmatclub.com/forum/is-triangle-abc-obtuse-angled-203020.html?hilit=acute
3) https://gmatclub.com/forum/are-all-angles-of-triangle-abc-smaller-than-90-degrees-129298.html?hilit=acute

Happy Learning

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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post Updated on: 12 Jul 2020, 04:15
1
Statement 1:
The two sides (AB and BC) and the included angle (Angle B) are given. We can deduce that the other two angles are fixed as the two sides are also fixed.
Sufficient.

Statement 2:
If the three sides A^2, B^2, C^2 together satisfy the pythagoras theorem then the triangle is a right angled triangle.
Hence, A^4 + B^4 = C^4

Let's assume
A^2 = 3
B^2 = 4
C^2 = 5
So,
3^2 + 4^2 = 5^2

Now,
A = \(\sqrt{3}\)
B =\(\sqrt{4}\)
C =\( \sqrt{5}\)

For a triangle to be acute, A^2 + B^2 > C^2
Here, 3 + 4 > 7
Thus, the given triangle is acute angled triangle.
Hence Sufficient.

Option D

Originally posted by NikuHBS on 11 Jul 2020, 18:26.
Last edited by NikuHBS on 12 Jul 2020, 04:15, edited 3 times in total.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 18:34
Is the triangle ABC with sides a, b, c an acute angles triangle?
1) The Sides AB (length = c) and BC (length = a) are 13 and 17 respectively and the angle ABC is 30 deg.
2) The triangle with sides a^2, b^2 and c^2 is a right-angled triangle.

1) using law of cosine,
c^2= a^2+b^2-2abcos(c)
we can find length and check if it is acute or not
sufficient

2) a,b,c is right angled
sufficient

Ans D
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post Updated on: 12 Jul 2020, 04:15
Statement 1:
The two sides (AB and BC) and the included angle (Angle B) are given. We can deduce that the other two angles are fixed as the two sides are also fixed.
Sufficient.

Statement 2:
If the three sides A^2, B^2, C^2 together satisfy the pythagoras theorem then the triangle is a right angled triangle.
Hence, A^4 + B^4 = C^4

Let's assume
A^2 = 3
B^2 = 4
C^2 = 5
So,
3^2 + 4^2 = 5^2

Now,
A = \(\sqrt{3}\)
B =\(\sqrt{4}\)
C =\( \sqrt{5}\)

For a triangle to be acute, A^2 + B^2 > C^2
Here, 3 + 4 > 7
Thus, the given triangle is acute angled triangle.
Hence Sufficient.

Option D

Originally posted by NikuHBS on 11 Jul 2020, 18:39.
Last edited by NikuHBS on 12 Jul 2020, 04:15, edited 1 time in total.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 18:40
Option B
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 18:44
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 19:06
IMO D

Is the triangle ABC with sides a, b, c an acute angles triangle?

1) The Sides AB (length = c) and BC (length = a) are 13 and 17 respectively and the angle ABC is 30 deg.

Lets draw the altitude AD from A

In triangle ABD, AB=13, Angle ABD=30
So, AD= 13 sin30= 6.5
BD=13cos30= 11.25

AlSo, in triangle ACD, tan Angle DAC= (13-11.25)/6.5=0.88

Therefore , 30< DAC < 45
So angle BAC= 60+DAC = > 90
Sufficient.

2) The triangle with sides a^2, b^2 and c^2 is a right-angled triangle.

Let a^2=3, b^2=4, C^2=5 Forms a right angle triangle

Now , From cosine rule, Cos angle ABC= [(c^2+a^2)-b^2]/2ac = 4/2sqrt15= 2/4=1/2 ~ 60 degree
Similarly, other angles can be found.

Sufficient.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 19:14
answer :D
State 1:sufficient
given AC=b,AB=c=13 and BC=a=17,Angle ABC=30
triangle ABC WILL be acute if BC^2<AB^2+AC^2
Now applying cos30=a^2+c^2-b^2/2ac
putting value and solving, we get b^2=458-221sqrt3= approx. 76
conditionBC^2>AB^2+AC^2 is satisfied as 289>13^2+76
State 2:sufficient
from given condition ,let c^2 be the largest side,then c^4=a^4+b^4=(a^2+b^2)^2-2 x a^2 x b^2
c^2<a^2+b^2
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 19:25
IMO E

First statement does not give us enough information about the angles of the triangle, there is a possibility that one of the anglkes can be more than 90 degree.
From the second statement alone we cant get the answer, knowing that a^2, b^2,c^2 makes a right angle triangle does not give us enough information if the triangle with a,b,c is a right angle triangle or not.

Using both statements we cannot conclude anything here.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 11 Jul 2020, 19:30
Is the triangle ABC with sides a, b, c an acute angles triangle?
1) The Sides AB (length = c) and BC (length = a) are 13 and 17 respectively and the angle ABC is 30 deg.
2) The triangle with sides a^2, b^2 and c^2 is a right-angled triangle.

Sum of angles of a triangle = 180

S1: Enough to know that sum of remaining angles will be 150. Could be 90+70 (not acute) or 75+75 (acute)
IS

S2: Square of the sides makes it Pythagoras and since one anglie is 90 deg the other two will add up to 90 deg i.e. their values will also be less than 90 deg.
Suffiecient

Answer is B.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 12 Jul 2020, 00:09
IMO D

For the solution, PFA
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 12 Jul 2020, 00:36
1- from statement side a is 4<a<30. Since angle C has to be less than 30 degree , been shorter than side b of angel B equals to 30 degree. Hence taking the max value of angle C +angle b = 29.99+30= 59.99. Angle A = 180-59.99= 120.01.
So not an acute angle triangle

2- Since Pythagoras triangle property is applicable, not an acute angle triangle .

Ans is D
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 12 Jul 2020, 01:23
only option 2 is sufficient
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Is the triangle ABC with sides a, b,  [#permalink]

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New post Updated on: 16 Jul 2020, 01:12
Awesome question
Acute triangle ?
1. Two sides and one included angle is given. Follow below method

Image

For this type of triangle, we must use The Law of Cosines first to calculate the third side of the triangle; then we can use The Law of Sines to find one of the other two angles, and finally use Angles of a Triangle to find the last angle.

Law of cosines:
Image
The Law of Cosines (or Cosine Rule) says:
c2 = a2 + b2 − 2ab cos(C)

Law of Sines: (refer above picture)
\(\frac{(a}{sin A)} = \frac{b}{sin B} = \frac{c}{sin C}\)

Angles of a Triangle:
Image
A + B + C = 180°

So, all in all statement 1 is sufficient to know the type of triangle

2. This means that \(a^2 b^2 and c^2\) form a Pythagorean triplet(there are many of them- (3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17) (9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21,29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97)

Square root of these triplet form the sides of the triangle. e.g. triplet - 3,4,5
corresponding sides-\( \sqrt{3}, 2 and \sqrt{5}\)

Now a triangle can be formed using - https://www.mathopenref.com/consttrianglesss.html

But point worth noting here is that - side is \(c ^2 = a^2 + b^2\) that means the angle between sides sides a and b is 90 degrees, but
This does not make us sure that the other angles are acute- refer picture below. Even if other angles are acute or obtuse atleast one angle is 90 degree(triangle is not acute)

Hence, Statement 2 is sufficient
Answer Option D
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Triangles side concept.jpg [ 119.93 KiB | Viewed 596 times ]


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Originally posted by aggvipul on 12 Jul 2020, 03:35.
Last edited by aggvipul on 16 Jul 2020, 01:12, edited 1 time in total.
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Re: Is the triangle ABC with sides a, b,  [#permalink]

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New post 12 Jul 2020, 04:34
Is the triangle ABC with sides a, b, c an acute angles triangle?
1) The Sides AB (length = c) and BC (length = a) are 13 and 17 respectively and the angle ABC is 73.8 deg.
2) The triangle with sides a^2, b^2 and c^2 is a right-angled triangle.

an acute ∆ ; a^2+b^2>c^2 is a<=b<=c
all angles are <90*
#1
The Sides AB (length = c) and BC (length = a) are 13 and 17 respectively and the angle ABC is 73.8 deg.
third side angle is 73.8 ; and sum of angle for length c & a ; 106.2 whose sum of can 90+16.2 or it can be 73 +33.2 we get yes and no ; insufficient
#2
The triangle with sides a^2, b^2 and c^2 is a right-angled triangle.
a right angled ∆ a^2+b^2=c^2
such that one of the angles is 90* ; so we can say that a, b, c is not an acute angles triangle.
OPTION B
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Re: Is the triangle ABC with sides a, b,   [#permalink] 12 Jul 2020, 04:34

Is the triangle ABC with sides a, b,

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