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In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of fourth triangle to that of seventh triangle?
- a)256 : 1
- b)128 : 1
- c)64 : 1
- d)16 : 1
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
In an equilateral triangle, another equilateral triangle is drawn insi...
Let the length of side of first equilateral triangle = x;
As shown in the figure, when an equilateral triangle is drawn inside another equilateral triangle joining the mid-points of the sides, the sides of that triangle will be half of the side of the 1st equilateral triangle;
In this way;
Side of 2nd triangle = x/2;
Side of 3rd triangle = x/4;
Side of 4th triangle = x/8;
Side of 5th triangle = x/16;
Side of 6th triangle = x/32;
Side of 7th triangle = x/64;
∴ Ratio of the area of 4th to 7th triangle = [√3/4 × (x/8)2] : [√3/4 × (x/64)2] = 64 : 1
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In an equilateral triangle, another equilateral triangle is drawn insi...
Problem:
In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of the fourth triangle to that of the seventh triangle?
Solution:
Step 1: Identify the given information
- We have an equilateral triangle.
- Another equilateral triangle is drawn inside it by joining the midpoints of the sides.
- This process is repeated up to seven times.
Step 2: Understand the problem
- We need to find the ratio of the area of the fourth triangle to that of the seventh triangle.
Step 3: Visualize the triangles
- Let's label the vertices of the outer equilateral triangle as A, B, and C.
- The midpoints of the sides of the outer triangle form another equilateral triangle inside, which we will label as P, Q, and R.
- Similarly, we can label the midpoints of the sides of each subsequent triangle as follows:
- 2nd triangle: D, E, and F
- 3rd triangle: G, H, and I
- 4th triangle: J, K, and L
- 5th triangle: M, N, and O
- 6th triangle: X, Y, and Z
- 7th triangle: W, V, and U
Step 4: Observe the pattern
- By observing the pattern, we can see that each triangle formed is similar to the outer equilateral triangle.
- The ratio of the sides of each triangle to the previous triangle is 1:2.
- Therefore, the ratio of the areas of each triangle to the previous triangle is (1:2)^2 = 1:4.
Step 5: Calculate the ratio
- The ratio of the area of the fourth triangle (J, K, L) to that of the seventh triangle (W, V, U) is (1:4)^3 = 1:64.
Step 6: Choose the correct option
- The correct answer is option 'c': 64:1.
In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of the fourth triangle to that of the seventh triangle?
Solution:
Step 1: Identify the given information
- We have an equilateral triangle.
- Another equilateral triangle is drawn inside it by joining the midpoints of the sides.
- This process is repeated up to seven times.
Step 2: Understand the problem
- We need to find the ratio of the area of the fourth triangle to that of the seventh triangle.
Step 3: Visualize the triangles
- Let's label the vertices of the outer equilateral triangle as A, B, and C.
- The midpoints of the sides of the outer triangle form another equilateral triangle inside, which we will label as P, Q, and R.
- Similarly, we can label the midpoints of the sides of each subsequent triangle as follows:
- 2nd triangle: D, E, and F
- 3rd triangle: G, H, and I
- 4th triangle: J, K, and L
- 5th triangle: M, N, and O
- 6th triangle: X, Y, and Z
- 7th triangle: W, V, and U
Step 4: Observe the pattern
- By observing the pattern, we can see that each triangle formed is similar to the outer equilateral triangle.
- The ratio of the sides of each triangle to the previous triangle is 1:2.
- Therefore, the ratio of the areas of each triangle to the previous triangle is (1:2)^2 = 1:4.
Step 5: Calculate the ratio
- The ratio of the area of the fourth triangle (J, K, L) to that of the seventh triangle (W, V, U) is (1:4)^3 = 1:64.
Step 6: Choose the correct option
- The correct answer is option 'c': 64:1.
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In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of fourth triangle to that of seventh triangle?a)256 : 1b)128 : 1c)64 : 1d)16 : 1Correct answer is option 'C'. Can you explain this answer?
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In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of fourth triangle to that of seventh triangle?a)256 : 1b)128 : 1c)64 : 1d)16 : 1Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of fourth triangle to that of seventh triangle?a)256 : 1b)128 : 1c)64 : 1d)16 : 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an equilateral triangle, another equilateral triangle is drawn inside joining the mid-points of the sides of the given equilateral triangle and the process continues up to seven times. What is the ratio of the area of fourth triangle to that of seventh triangle?a)256 : 1b)128 : 1c)64 : 1d)16 : 1Correct answer is option 'C'. Can you explain this answer?.
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