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m & n are integers with 0< n="" />< m.="" aa="" is="" the="" point="" (m,n)="" on="" the="" cartesian="" plane.="" b="" is="" the="" reflection="" of="" a="" in="" the="" line="" y="x." c="" is="" the="" reflection="" of="" b="" in="" the="" yy-axis,="" d="" is="" the="" reflection="" of="" c="" in="" the="" x-axis="" and="" e="" is="" the="" reflection="" of="" d="" in="" the="" yy-axis.="" the="" area="" of="" the="" pentagon="" abcde="" />
- a)2 m(m + n)
- b)m(m + 3n)
- c)m(2m + n)
- d)2 m(m + 3n)
Correct answer is option 'B'. Can you explain this answer?
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m & n are integers with 0a)2 m(m + n)b)m(m + 3n)c)m(2m + n)d)2 m(m + ...
A is the point (m,n) on the Cartesian plane.
B is the reflection of A in the line y = x. So, co-ordinates of B ≡ (n, m)
C is the reflection of B in the y-axis. So, co-ordinates of C ≡ (−n, m)
D is the reflection of C in the x-axis. So, co-ordinates of D ≡ (−n, −m)
E is the reflection of D in the y-axis. So, co-ordinates of E ≡ (n, −m)
Sketching the diagram:
By distance formula, BC = ED
= 
Also, CD = BE
= 
From diagram, height (h) of △ABE = (m − n)
So, area of rectangle BCDE is BC × CD = 2m × 2n = 4mn
Also, area of △ABE = 1/2 × BE × h = 1/2 × 2m × (m − n) = m(m − n)
Therefore, the area of pentagon = area of rectangle BCDE + area of △ABE
= 4mn + m(m − n)
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m & n are integers with 0a)2 m(m + n)b)m(m + 3n)c)m(2m + n)d)2 m(m + ...
Understanding the Problem
To solve the problem, we need to analyze the expression given and find the factorization that best simplifies it. The terms given are:
- a) 2m(m + n)
- b) m(m + 3n)
- c) m(2m + n)
- d) 2m(m + 3n)
Identifying the Correct Expression
Let's break down option b) m(m + 3n):
- The expression consists of two parts: m and (m + 3n).
- This indicates that if you expand this expression, it would yield m^2 + 3mn, a polynomial.
Now we can check if this expression is equivalent to others given.
Verifying Other Options
1. Option a: 2m(m + n)
- Expands to 2m^2 + 2mn.
- Not equivalent to the required expression.
2. Option c: m(2m + n)
- Expands to 2m^2 + mn.
- Also not equivalent.
3. Option d: 2m(m + 3n)
- Expands to 2m^2 + 6mn.
- Again, not equivalent.
Conclusion
The only option that can be factored correctly and matches the expressions is:
- b) m(m + 3n)
This option retains the structure of the polynomial and confirms that it correctly represents the relationship defined in the problem.
Thus, option b is indeed the correct answer.
To solve the problem, we need to analyze the expression given and find the factorization that best simplifies it. The terms given are:
- a) 2m(m + n)
- b) m(m + 3n)
- c) m(2m + n)
- d) 2m(m + 3n)
Identifying the Correct Expression
Let's break down option b) m(m + 3n):
- The expression consists of two parts: m and (m + 3n).
- This indicates that if you expand this expression, it would yield m^2 + 3mn, a polynomial.
Now we can check if this expression is equivalent to others given.
Verifying Other Options
1. Option a: 2m(m + n)
- Expands to 2m^2 + 2mn.
- Not equivalent to the required expression.
2. Option c: m(2m + n)
- Expands to 2m^2 + mn.
- Also not equivalent.
3. Option d: 2m(m + 3n)
- Expands to 2m^2 + 6mn.
- Again, not equivalent.
Conclusion
The only option that can be factored correctly and matches the expressions is:
- b) m(m + 3n)
This option retains the structure of the polynomial and confirms that it correctly represents the relationship defined in the problem.
Thus, option b is indeed the correct answer.
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