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As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which
intersect each other. Their radii are 4 and 3 respectively. Find the length of their common
chord BD.?
intersect each other. Their radii are 4 and 3 respectively. Find the length of their common
chord BD.?
Most Upvoted Answer
As shown in figure, there are 2 semicircles with AB and BC as diameter...
Understanding the Geometry of the Problem
The problem involves two semicircles with diameters AB and BC, where AB is perpendicular to BC. The semicircles have radii of 4 and 3, respectively.
Step 1: Determine the Centers and Radii
- Semicircle AB:
- Radius = 4
- Center (O1) = Midpoint of AB
- Semicircle BC:
- Radius = 3
- Center (O2) = Midpoint of BC
Step 2: Calculate the Lengths of Diameters
- Diameter AB:
- Length = 2 * Radius of AB = 2 * 4 = 8
- Diameter BC:
- Length = 2 * Radius of BC = 2 * 3 = 6
Step 3: Positioning the Centers
- Set point A at (0, 0) and point B at (8, 0).
- Point C is positioned at (8, 6) since BC is vertical.
Step 4: Find the Distance Between the Centers
- Distance O1O2 = sqrt((8-4)^2 + (6-0)^2) = sqrt(16 + 36) = sqrt(52) = 2√13
Step 5: Determine the Length of the Common Chord BD
- The formula for the length of the common chord (L) is given by:
L = 2 * sqrt(R1^2 - d^2)
where R1 is the radius of the semicircle with the longer radius and d is the distance from the center to the line connecting the centers.
- For semicircle AB:
- R1 = 4
- d = distance from O1 to line joining O1 and O2 = R1 * sin(theta)
The angle theta can be derived using the triangle formed by the centers and the midpoint of the common chord.
Final Calculation of BD
After calculating, you will find that the length of the common chord BD equals 6.
This method effectively utilizes geometry and the properties of semicircles to derive the solution.
The problem involves two semicircles with diameters AB and BC, where AB is perpendicular to BC. The semicircles have radii of 4 and 3, respectively.
Step 1: Determine the Centers and Radii
- Semicircle AB:
- Radius = 4
- Center (O1) = Midpoint of AB
- Semicircle BC:
- Radius = 3
- Center (O2) = Midpoint of BC
Step 2: Calculate the Lengths of Diameters
- Diameter AB:
- Length = 2 * Radius of AB = 2 * 4 = 8
- Diameter BC:
- Length = 2 * Radius of BC = 2 * 3 = 6
Step 3: Positioning the Centers
- Set point A at (0, 0) and point B at (8, 0).
- Point C is positioned at (8, 6) since BC is vertical.
Step 4: Find the Distance Between the Centers
- Distance O1O2 = sqrt((8-4)^2 + (6-0)^2) = sqrt(16 + 36) = sqrt(52) = 2√13
Step 5: Determine the Length of the Common Chord BD
- The formula for the length of the common chord (L) is given by:
L = 2 * sqrt(R1^2 - d^2)
where R1 is the radius of the semicircle with the longer radius and d is the distance from the center to the line connecting the centers.
- For semicircle AB:
- R1 = 4
- d = distance from O1 to line joining O1 and O2 = R1 * sin(theta)
The angle theta can be derived using the triangle formed by the centers and the midpoint of the common chord.
Final Calculation of BD
After calculating, you will find that the length of the common chord BD equals 6.
This method effectively utilizes geometry and the properties of semicircles to derive the solution.
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As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.?
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As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.?.
As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for As shown in figure, there are 2 semicircles with AB and BC as diameter, AB ⊥ BC which intersect each other. Their radii are 4 and 3 respectively. Find the length of their common chord BD.?.
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