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ABC and BDE are two equilateral triangles such that D is mid-point of BC. Ratio of the areas of triangles ABC and BDE is
- a)2 : 1
- b)1 : 4
- c)1 : 2
- d)4 : 1
Correct answer is option 'D'. Can you explain this answer?
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To find the ratio of the areas of triangles ABC and BDE, we need to compare their areas.
Given:
- ABC and BDE are equilateral triangles.
- D is the midpoint of BC.
Let's start by finding the relationship between the sides of the two triangles.
1. Relationship between the sides:
- In an equilateral triangle, all sides are equal.
- Therefore, AB = BC = CA and BD = DE = EB.
Now, let's calculate the ratio of the areas of the two triangles.
2. Area of triangle ABC:
- The area of an equilateral triangle can be found using the formula: (sqrt(3)/4) * side^2.
- Let's assume the side length of ABC is 's'.
- Therefore, the area of ABC = (sqrt(3)/4) * s^2.
3. Area of triangle BDE:
- Since BD = DE = EB, triangle BDE is also an equilateral triangle.
- Let's assume the side length of BDE is 'x'.
- Therefore, the area of BDE = (sqrt(3)/4) * x^2.
4. Relationship between the side lengths:
- From step 1, we know that AB = BC = CA and BD = DE = EB.
- Since D is the midpoint of BC, we can conclude that BD = DC.
- Therefore, BD + DC = BC.
- Substituting the values, we get x + x = s.
- Simplifying, we have 2x = s.
- So, x = s/2.
5. Substituting the values in the area formulas:
- Area of ABC = (sqrt(3)/4) * s^2 = (sqrt(3)/4) * (2x)^2 = (sqrt(3)/4) * 4x^2 = sqrt(3) * x^2.
- Area of BDE = (sqrt(3)/4) * x^2.
6. Finding the ratio:
- Ratio of the areas = (Area of ABC) : (Area of BDE)
= (sqrt(3) * x^2) : ((sqrt(3)/4) * x^2)
= (sqrt(3) * x^2) / ((sqrt(3)/4) * x^2)
= (sqrt(3) * x^2) * (4/sqrt(3) * x^2)
= 4/1
= 4 : 1.
Therefore, the correct answer is option 'D', which states that the ratio of the areas of triangles ABC and BDE is 4 : 1.
Given:
- ABC and BDE are equilateral triangles.
- D is the midpoint of BC.
Let's start by finding the relationship between the sides of the two triangles.
1. Relationship between the sides:
- In an equilateral triangle, all sides are equal.
- Therefore, AB = BC = CA and BD = DE = EB.
Now, let's calculate the ratio of the areas of the two triangles.
2. Area of triangle ABC:
- The area of an equilateral triangle can be found using the formula: (sqrt(3)/4) * side^2.
- Let's assume the side length of ABC is 's'.
- Therefore, the area of ABC = (sqrt(3)/4) * s^2.
3. Area of triangle BDE:
- Since BD = DE = EB, triangle BDE is also an equilateral triangle.
- Let's assume the side length of BDE is 'x'.
- Therefore, the area of BDE = (sqrt(3)/4) * x^2.
4. Relationship between the side lengths:
- From step 1, we know that AB = BC = CA and BD = DE = EB.
- Since D is the midpoint of BC, we can conclude that BD = DC.
- Therefore, BD + DC = BC.
- Substituting the values, we get x + x = s.
- Simplifying, we have 2x = s.
- So, x = s/2.
5. Substituting the values in the area formulas:
- Area of ABC = (sqrt(3)/4) * s^2 = (sqrt(3)/4) * (2x)^2 = (sqrt(3)/4) * 4x^2 = sqrt(3) * x^2.
- Area of BDE = (sqrt(3)/4) * x^2.
6. Finding the ratio:
- Ratio of the areas = (Area of ABC) : (Area of BDE)
= (sqrt(3) * x^2) : ((sqrt(3)/4) * x^2)
= (sqrt(3) * x^2) / ((sqrt(3)/4) * x^2)
= (sqrt(3) * x^2) * (4/sqrt(3) * x^2)
= 4/1
= 4 : 1.
Therefore, the correct answer is option 'D', which states that the ratio of the areas of triangles ABC and BDE is 4 : 1.
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ABC and BDE are two equilateral triangles such that D is mid-point of BC. Ratio of the areas of triangles ABC and BDE isa)2 : 1b)1 : 4c)1 : 2d)4 : 1Correct answer is option 'D'. Can you explain this answer?
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ABC and BDE are two equilateral triangles such that D is mid-point of BC. Ratio of the areas of triangles ABC and BDE isa)2 : 1b)1 : 4c)1 : 2d)4 : 1Correct answer is option 'D'. Can you explain this answer? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about ABC and BDE are two equilateral triangles such that D is mid-point of BC. Ratio of the areas of triangles ABC and BDE isa)2 : 1b)1 : 4c)1 : 2d)4 : 1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ABC and BDE are two equilateral triangles such that D is mid-point of BC. Ratio of the areas of triangles ABC and BDE isa)2 : 1b)1 : 4c)1 : 2d)4 : 1Correct answer is option 'D'. Can you explain this answer?.
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