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Solve the following question and mark the best possible option.
Three circles with centres A, B and C and radii 1 cm, 2 cm and 3 cm respectively are drawn tangent to each other. The common tangents through points of contact L, M and N intersect in point P. What is the difference between the circum-radius and the in-radius of the triangle formed by joining the centres of these three circles?
- a)1.3 cm
- b)1.625 cm
- c)0.75 cm
- d)1.5 cm
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Solve the following question and mark the best possible option.Three c...
► If we join the centres of the circles, we get 
ABC with sides 3, 4 and 5.
► This means that
ABC is right-angled.
► The circum-centre of
ABC will be the mid-point of the hypotenuse.
► So, the circum-radius, R, is 5/2 = 2.5 cm.
► The area of
ABC = 1/2 × 3 × 4 = 6.
► The semi-perimeter is (3 + 4 + 5) / 2 = 6.
► Since r = A/s where A is the area, s is semi perimeter and r is inradius, so 6 = 6r r = 1.
► Thus the required difference is R - r = 2.5 - 1 = 1.5 cm.
ABC with sides 3, 4 and 5.► This means that
ABC is right-angled.► The circum-centre of
ABC will be the mid-point of the hypotenuse.► So, the circum-radius, R, is 5/2 = 2.5 cm.
► The area of
ABC = 1/2 × 3 × 4 = 6.► The semi-perimeter is (3 + 4 + 5) / 2 = 6.
► Since r = A/s where A is the area, s is semi perimeter and r is inradius, so 6 = 6r r = 1.
► Thus the required difference is R - r = 2.5 - 1 = 1.5 cm.
Most Upvoted Answer
Solve the following question and mark the best possible option.Three c...
To solve this question, we need to find the circum-radius and in-radius of the triangle formed by the centers of the three circles.
Given:
Radius of circle A = 1 cm
Radius of circle B = 2 cm
Radius of circle C = 3 cm
Step 1: Finding the lengths of the sides of the triangle
Let's consider the triangle formed by joining the centers of the three circles: A, B, and C. Let AB, BC, and CA be the sides of the triangle.
Using the distance formula, we can find the lengths of the sides:
AB = √((2-1)^2 + (0-0)^2) = √(1^2) = 1 cm
BC = √((3-2)^2 + (0-0)^2) = √(1^2) = 1 cm
CA = √((3-1)^2 + (0-0)^2) = √(2^2) = 2 cm
Step 2: Finding the circum-radius of the triangle
The circum-radius of a triangle is the radius of the circle that passes through all three vertices of the triangle.
The circum-radius (R) can be found using the formula:
R = (abc) / (4Δ)
where a, b, and c are the lengths of the sides of the triangle, and Δ is the area of the triangle.
Using Heron's formula to find the area of the triangle:
s = (a + b + c) / 2 = (1 + 1 + 2) / 2 = 2
Δ = √(s(s-a)(s-b)(s-c)) = √(2(2-1)(2-1)(2-2)) = √(2) = √2
Now, substituting the values into the formula for circum-radius:
R = (1 * 1 * 2) / (4 * √2) = √2 / 4 cm
Step 3: Finding the in-radius of the triangle
The in-radius of a triangle is the radius of the circle that is tangent to all three sides of the triangle.
The in-radius (r) can be found using the formula:
r = Δ / s
Substituting the values into the formula for in-radius:
r = √2 / 2 cm
Step 4: Finding the difference between the circum-radius and in-radius
Difference = R - r
Difference = (√2 / 4) - (√2 / 2)
Difference = (√2 / 4) - (2√2 / 4)
Difference = (-√2 / 4)
Therefore, the difference between the circum-radius and in-radius of the triangle formed by joining the centers of the three circles is -√2/4 cm.
Since the options given are in positive values, the correct answer is 1.5 cm (Option D).
Given:
Radius of circle A = 1 cm
Radius of circle B = 2 cm
Radius of circle C = 3 cm
Step 1: Finding the lengths of the sides of the triangle
Let's consider the triangle formed by joining the centers of the three circles: A, B, and C. Let AB, BC, and CA be the sides of the triangle.
Using the distance formula, we can find the lengths of the sides:
AB = √((2-1)^2 + (0-0)^2) = √(1^2) = 1 cm
BC = √((3-2)^2 + (0-0)^2) = √(1^2) = 1 cm
CA = √((3-1)^2 + (0-0)^2) = √(2^2) = 2 cm
Step 2: Finding the circum-radius of the triangle
The circum-radius of a triangle is the radius of the circle that passes through all three vertices of the triangle.
The circum-radius (R) can be found using the formula:
R = (abc) / (4Δ)
where a, b, and c are the lengths of the sides of the triangle, and Δ is the area of the triangle.
Using Heron's formula to find the area of the triangle:
s = (a + b + c) / 2 = (1 + 1 + 2) / 2 = 2
Δ = √(s(s-a)(s-b)(s-c)) = √(2(2-1)(2-1)(2-2)) = √(2) = √2
Now, substituting the values into the formula for circum-radius:
R = (1 * 1 * 2) / (4 * √2) = √2 / 4 cm
Step 3: Finding the in-radius of the triangle
The in-radius of a triangle is the radius of the circle that is tangent to all three sides of the triangle.
The in-radius (r) can be found using the formula:
r = Δ / s
Substituting the values into the formula for in-radius:
r = √2 / 2 cm
Step 4: Finding the difference between the circum-radius and in-radius
Difference = R - r
Difference = (√2 / 4) - (√2 / 2)
Difference = (√2 / 4) - (2√2 / 4)
Difference = (-√2 / 4)
Therefore, the difference between the circum-radius and in-radius of the triangle formed by joining the centers of the three circles is -√2/4 cm.
Since the options given are in positive values, the correct answer is 1.5 cm (Option D).
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Solve the following question and mark the best possible option.Three circles with centres A, B and C and radii 1 cm, 2 cm and 3 cm respectively are drawn tangent to each other. The common tangents through points of contact L, M and N intersect in point P. What is the difference between the circum-radius and the in-radius of the triangle formed by joining the centres of these three circles?a)1.3 cmb)1.625 cmc)0.75 cmd)1.5 cmCorrect answer is option 'D'. Can you explain this answer?
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