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If the smallest angle of a triangle measures 45 degrees, which of the

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Bunuel
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If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   28 May 2019, 01:19
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If the smallest angle of a triangle measures 45 degrees, which of the following could be the ratio of the three angles of the triangle?

I. 2:3:3
II. 3:4:5
III. 9:10:17

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III
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E
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Re: If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   28 May 2019, 01:29
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Bunuel wrote:
If the smallest angle of a triangle measures 45 degrees, which of the following could be the ratio of the three angles of the triangle?

I. 2:3:3
II. 3:4:5
III. 9:10:17

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


I 45 67.5 67.5
II 45 60 75
III 45 50 85

All I II and III E
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If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   Updated on: 30 Jan 2022, 15:12
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A combination of Ratio and Geometry

Given : Smallest angle of triangle = 45 degree

To find : Could be ratio of angle of triangle whose smallest angle is 45 degree

I. 2:3:3
II. 3:4:5
III. 9:10:17

Considering case I :

2 = 45 deg (i.e 1 = 22.5 deg)

So, similarly - 4 = 4*22.5 = 67.5 deg
and - 4 = 4*22.5 = 67.5 deg

adding all three angles >> 45+67.5+67.5 = 180 degree
So case I. is a possible ratio as sum derived is equal to 180 (property of triangle : sum of all angles =180)

Similar for case II : angles comes to be 45:60:75 (sum comes to be 180)
Similar for case III : angles comes to be 45:50:85 (sum comes to be 180)

So all 3 cases are possible .... Ans E
----------------------

Originally posted by AOBang on 18 Jul 2020, 21:38.
Last edited by Bunuel on 30 Jan 2022, 15:12, edited 5 times in total.
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Re: If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   18 Jul 2020, 21:47
Expert Reply
Bunuel wrote:
If the smallest angle of a triangle measures 45 degrees, which of the following could be the ratio of the three angles of the triangle?

I. 2:3:3
II. 3:4:5
III. 9:10:17

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


I. 2:3:3 => \(2x:3x:3x....2x=45...x=22.5\), so \(2x+3x+3x=8x=8*22.5=180\)..YES
II. 3:4:5 => \(3x:4x:5x....3x=45...x=15\), so \(3x+4x+5x=12x=12*15=180\)..YES
III. 9:10:17=> \(9x:10x:17x....9x=45...x=5\), so \(9x+10x+17x=36x=36*5=180\)..YES

E
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If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   17 Aug 2020, 21:27
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If the Smallest Angle of the Triangle = 45 Degrees, then it will also be the Smallest "PART" in the Ratio.

For Each Answer Choice check if: (Smallest PART of Ratio) / (WHOLE Ratio) = 45 / 180

if the Fractions Reduce to be Equal, then it is a Valid Triangle

I. Is 2 / (2 + 3 + 3) = 45 / 180 ?

45/180 = 1/4 2/8 = 1/4 YES

II. Is 3 / (3 + 4 + 5) = 45 / 180 or 1/4?

3/12 = 1/4 YES

III. Is 9 / (9 + 10 + 17) = 45 / 180 or 1/4?

9/36 = 1/4 YES

Answer Choice E - All 3 could be Valid Ratios of the Angles of a Triangle if the smallest Angle = 45 degrees
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Re: If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   18 Aug 2020, 01:19
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Sum of all the angles ina triangle is \(180^o\)

Given: Smallest angle = \(45^o\)

Then, the sum of the remaining two angles: \(135^o\)

Option I: Smallest angle (45/2) = \(22.5^0\)

- Other angles: (3+3 = 6) : 135 / 6 = 22.5 Ratio is 22.5 * 2 : 67.5 * 3 : 67.5 * 3 (45 + 135 = 180) - Condition I TRUE

Option II: Smallest angle (45/3) = \(15^0\)

- Other angles: (4+5 = 9) : 135 / 9 = 15 Ratio is 15 * 3 : 15 * 4 : 15 * 5 (45 + 135 = 180)- Condition II TRUE

Option III: Smallest angle (45/9) = \(5^0\)

- Other angles: (10+17 = 27) : 135 / 27 = 5 Ratio is 5 * 9 : 5 * 10 : 5 * 17 (45 + 135 = 180)- Condition III TRUE

Answer E
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Re: If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   28 Aug 2020, 20:44
So, I solved it using a different way. In a triangle, sum of 2 sides needs to be greater than the 3rd side. So basically if you can prove that , 2 short sides> Longest side, then it is a possible triangle. Hence
1. 2,3,3 = 2+3 >5 Yes, possible
2. 3,4,5 = 3+4>5 Yes, possible
3. 9,10,17 =9+10>17 Yes, possible
Hence all the 3 cases are possible. Hence option E
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Re: If the smallest angle of a triangle measures 45 degrees, which of the [#permalink] New post   30 Jan 2022, 15:04
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Re: If the smallest angle of a triangle measures 45 degrees, which of the [#permalink]
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