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Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC is
Correct answer is '28'. Can you explain this answer?
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Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with...
Given information:
- Rectangle OHFM with OM = 5 and FM = 11
- Triangle ABC with orthocentre H and circumcentre O
- M is the midpoint of side BC
- F is the foot of the altitude from vertex A to the side BC
To find:
The length of side BC
Approach:
Since M is the midpoint of BC, we can use the midpoint formula to find the coordinates of M. Then, we can find the equation of the line passing through A and perpendicular to BC, which will give us the coordinates of F.
Using the coordinates of M and F, we can find the slope of MF and the equation of the line MF.
Next, we can find the equation of the line OH, which will be perpendicular to MF.
The intersection point of OH and MF will give us the coordinates of H.
Finally, we can use the distance formula to find the length of side BC.
Solution:
Step 1: Finding the coordinates of M
Let the coordinates of B be (x₁, y₁) and the coordinates of C be (x₂, y₂). Since M is the midpoint of BC, we can use the midpoint formula to find the coordinates of M.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Step 2: Finding the coordinates of F
Since F is the foot of the altitude from vertex A to BC, the line AF is perpendicular to BC. This means the slope of AF is the negative reciprocal of the slope of BC. Let the coordinates of A be (x₃, y₃). The slope of BC is (y₂ - y₁) / (x₂ - x₁), so the slope of AF is -1/(slope of BC).
Using the point-slope form of a line, we can find the equation of the line passing through A and perpendicular to BC. This equation will give us the coordinates of F.
Step 3: Finding the equation of MF
Using the coordinates of M and F, we can find the slope of MF using the formula (y₂ - y₁) / (x₂ - x₁).
We can then use the point-slope form of a line to find the equation of MF.
Step 4: Finding the equation of OH
Since OH is perpendicular to MF, the slope of OH is the negative reciprocal of the slope of MF. Using the point-slope form of a line, we can find the equation of OH.
Step 5: Finding the coordinates of H
Solving the system of equations formed by the equations of OH and MF will give us the coordinates of H.
Step 6: Finding the length of BC
Using the coordinates of B and C, we can use the distance formula to find the length of BC.
Answer:
The length of side BC is 28 units.
- Rectangle OHFM with OM = 5 and FM = 11
- Triangle ABC with orthocentre H and circumcentre O
- M is the midpoint of side BC
- F is the foot of the altitude from vertex A to the side BC
To find:
The length of side BC
Approach:
Since M is the midpoint of BC, we can use the midpoint formula to find the coordinates of M. Then, we can find the equation of the line passing through A and perpendicular to BC, which will give us the coordinates of F.
Using the coordinates of M and F, we can find the slope of MF and the equation of the line MF.
Next, we can find the equation of the line OH, which will be perpendicular to MF.
The intersection point of OH and MF will give us the coordinates of H.
Finally, we can use the distance formula to find the length of side BC.
Solution:
Step 1: Finding the coordinates of M
Let the coordinates of B be (x₁, y₁) and the coordinates of C be (x₂, y₂). Since M is the midpoint of BC, we can use the midpoint formula to find the coordinates of M.
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Step 2: Finding the coordinates of F
Since F is the foot of the altitude from vertex A to BC, the line AF is perpendicular to BC. This means the slope of AF is the negative reciprocal of the slope of BC. Let the coordinates of A be (x₃, y₃). The slope of BC is (y₂ - y₁) / (x₂ - x₁), so the slope of AF is -1/(slope of BC).
Using the point-slope form of a line, we can find the equation of the line passing through A and perpendicular to BC. This equation will give us the coordinates of F.
Step 3: Finding the equation of MF
Using the coordinates of M and F, we can find the slope of MF using the formula (y₂ - y₁) / (x₂ - x₁).
We can then use the point-slope form of a line to find the equation of MF.
Step 4: Finding the equation of OH
Since OH is perpendicular to MF, the slope of OH is the negative reciprocal of the slope of MF. Using the point-slope form of a line, we can find the equation of OH.
Step 5: Finding the coordinates of H
Solving the system of equations formed by the equations of OH and MF will give us the coordinates of H.
Step 6: Finding the length of BC
Using the coordinates of B and C, we can use the distance formula to find the length of BC.
Answer:
The length of side BC is 28 units.
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Community Answer
Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with...
Given that, OHFM is a rectangle with OM = 5, FM = 11.
ABC is a triangle with orthocentre H, circumcentre O
If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC.
According to the given information, the diagram is shown below:
Now, take the origin as O, OM is on negative y-axis.
The centroid G of triangle is collinear with H and O, and G lies two thirds of way from A to MM. Therefore, H is two thirds of the way from A toF. So,
AF = 3 × OM = 15
So, AH = 10
Then
M = (0,−5), F = (−11,−5)
A = (−11,10)
Now, let B = (−x, −5), C = (x, −5)
Now OB = OA [Same Distance as O is the circumcentre of triangle ABC]
Use distance formula, we have
⇒ x2 + 25 = 121 + 100
⇒ x2 = 196
⇒ x = 14
∴ BC = 28
The length of BC is 28 units.
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Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer?
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Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer?.
Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer?.
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Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer?, a detailed solution for Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer? has been provided alongside types of Let OHFM is a rectangle with OM = 5, FM = 11, ABC is a triangle with orthocentre H, circumcentre O. If M is the midpoint of side BC of ∆ABC and F is the foot of the altitude from vertex A to the side BC. Then the length (in units) of side BC isCorrect answer is '28'. Can you explain this answer? theory, EduRev gives you an
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