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The angles of a triangle are in the ratio 3:5:10. Then the ratio of the smallest side to the greatest side is
- a)1: sin 10o
- b)1:2 sin 10o
- c)1: cos 10o
- d)1:2 cos10o
Correct answer is option 'D'. Can you explain this answer?
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The angles of a triangle are in the ratio 3:5:10. Then the ratio of th...
Given, the angles of a triangle are in the ratio 3:5:10.
Let the angles be 3x, 5x and 10x.
Sum of angles of a triangle = 180°
Therefore, 3x + 5x + 10x = 180°
⇒ 18x = 180°
⇒ x = 10°
Hence, the angles are 30°, 50° and 100°.
Ratio of sides in a triangle:
The ratio of the sides opposite to the angles of a triangle is directly proportional to the sine of the respective angle.
Let the sides opposite to the angles 30°, 50° and 100° be a, b and c, respectively.
Then, a:b:c = sin 30° : sin 50° : sin 100°
Finding the ratio of smallest side to the greatest side:
We need to find the ratio of the smallest side to the greatest side, which is a:c.
a:c = sin 30° : sin 100°
Using the trigonometric identity, sin(90° - θ) = cos θ, we can write:
sin 100° = sin(90° + 10°) = cos 10°
Hence, a:c = sin 30° : cos 10°
Using the trigonometric identity, sin 30° = 1/2 and cos 10° = √3/2 - sin 10°, we can simplify to get:
a:c = 1 : 2cos10°
Therefore, option (d) is the correct answer.
Let the angles be 3x, 5x and 10x.
Sum of angles of a triangle = 180°
Therefore, 3x + 5x + 10x = 180°
⇒ 18x = 180°
⇒ x = 10°
Hence, the angles are 30°, 50° and 100°.
Ratio of sides in a triangle:
The ratio of the sides opposite to the angles of a triangle is directly proportional to the sine of the respective angle.
Let the sides opposite to the angles 30°, 50° and 100° be a, b and c, respectively.
Then, a:b:c = sin 30° : sin 50° : sin 100°
Finding the ratio of smallest side to the greatest side:
We need to find the ratio of the smallest side to the greatest side, which is a:c.
a:c = sin 30° : sin 100°
Using the trigonometric identity, sin(90° - θ) = cos θ, we can write:
sin 100° = sin(90° + 10°) = cos 10°
Hence, a:c = sin 30° : cos 10°
Using the trigonometric identity, sin 30° = 1/2 and cos 10° = √3/2 - sin 10°, we can simplify to get:
a:c = 1 : 2cos10°
Therefore, option (d) is the correct answer.
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The angles of a triangle are in the ratio 3:5:10. Then the ratio of the smallest side to the greatest side isa)1: sin 10ob)1:2 sin 10oc)1: cos 10od)1:2 cos10oCorrect answer is option 'D'. Can you explain this answer?
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