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If the measures of the angles of a triangle are in the ratio. 1 ∶ 2 ∶ 3 and if the length of the smallest side of the triangle is 10 cm, then the length of the longest side is
- a)20 cm
- b)25 cm
- c)30 cm
- d)35 cm
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If the measures of the angles of a triangle are in the ratio. 1 ∶...
Let the angles of the triangles are in the ratio x ∶ 2x ∶ 3x
We know that sum of the angles in a triangle = 180°
⇒ x + 2x + 3x = 180
⇒ 6x = 180°
⇒ x = 30°
by sin formula sinA/a = sinB/b = sinC/c
∴ the angles of the triangle are 30°, 60° and 90°, so the sides are in the ratio of 1 ∶ 2 ∶ √3
∴ smallest side will be opposite to 30°
Let the sides of the triangle are in the ratio of x : 2x : √3x
But given smallest side x = 10 cm
⇒ remaining sides are 2x = 2(10) = 20 cm and √3x = √3(10) = 17.32 cm
∴ the longest side = 20 cm
We know that sum of the angles in a triangle = 180°
⇒ x + 2x + 3x = 180
⇒ 6x = 180°
⇒ x = 30°
by sin formula sinA/a = sinB/b = sinC/c
∴ the angles of the triangle are 30°, 60° and 90°, so the sides are in the ratio of 1 ∶ 2 ∶ √3
∴ smallest side will be opposite to 30°
Let the sides of the triangle are in the ratio of x : 2x : √3x
But given smallest side x = 10 cm
⇒ remaining sides are 2x = 2(10) = 20 cm and √3x = √3(10) = 17.32 cm
∴ the longest side = 20 cm
Most Upvoted Answer
If the measures of the angles of a triangle are in the ratio. 1 ∶...
Given Information:
The measures of the angles of a triangle are in the ratio 1:2:3.
The length of the smallest side of the triangle is 10 cm.
Explanation:
To solve this problem, we can use the fact that the measures of the angles in a triangle add up to 180 degrees. Let's assume the measures of the angles are x, 2x, and 3x.
Step 1: Finding the Measures of the Angles
Since the measures of the angles are in the ratio 1:2:3, we can write the equation:
x + 2x + 3x = 180
6x = 180
x = 30
Therefore, the measures of the angles are 30 degrees, 60 degrees, and 90 degrees.
Step 2: Finding the Length of the Other Sides
Since we know the measures of the angles, we can use trigonometry to find the lengths of the other sides. In a right-angled triangle, the sides are related by the trigonometric ratios.
Let's label the sides of the triangle as follows:
- The smallest side (opposite the smallest angle) is 10 cm.
- The side opposite the 60-degree angle is the hypotenuse.
- The side opposite the 90-degree angle is the longest side.
Using the Trigonometric Ratios:
- For the 60-degree angle: sin(60) = opposite/hypotenuse
- For the 90-degree angle: sin(90) = opposite/hypotenuse
Since sin(60) = √3/2 and sin(90) = 1, we can write:
√3/2 = 10/hypotenuse
1 = 10/longest side
Calculating the Length of the Longest Side:
From the equation 1 = 10/longest side, we can find the length of the longest side:
longest side = 10/1
longest side = 10 cm
Answer:
Therefore, the length of the longest side of the triangle is 10 cm, which corresponds to option A.
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If the measures of the angles of a triangle are in the ratio. 1 ∶ 2 ∶ 3 and if the length of the smallest side of the triangle is 10 cm, then the length of the longest side isa)20 cmb)25 cmc)30 cmd)35 cmCorrect answer is option 'A'. Can you explain this answer?
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