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In a ΔABC, if AB + BC = 10 cm, BC + CA = 12 cm, CA + AB = 16 cm, then the perimeter of the triangle
is :-
- a)19 cm
- b)17 cm
- c)38 cm
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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In a ΔABC, if AB + BC = 10 cm, BC + CA = 12 cm, CA + AB = 16 cm,...
Understanding the Problem
In triangle ABC, we are given three equations involving the sides:
- AB + BC = 10 cm
- BC + CA = 12 cm
- CA + AB = 16 cm
Let’s denote the sides as follows:
- AB = a
- BC = b
- CA = c
Now, we can rewrite the equations based on these variables:
1. a + b = 10 (Equation 1)
2. b + c = 12 (Equation 2)
3. c + a = 16 (Equation 3)
Finding the Sides
To find the individual lengths of the sides, we can manipulate these equations:
- From Equation 1: a = 10 - b
- Substitute a in Equation 3: c + (10 - b) = 16
- This simplifies to c - b = 6 (Equation 4)
Next, we express c from Equation 2:
- From Equation 2: c = 12 - b (Equation 5)
Now we can substitute Equation 5 into Equation 4:
- (12 - b) - b = 6
- 12 - 2b = 6
- 2b = 6
- b = 3 cm
Calculating the Other Sides
Now that we have b, we can find a and c:
- Substitute b back into Equation 1:
- a + 3 = 10 → a = 7 cm
- Substitute b into Equation 5:
- c = 12 - 3 = 9 cm
Calculating the Perimeter
Now, we can find the perimeter P of triangle ABC:
- P = a + b + c
- P = 7 cm + 3 cm + 9 cm
- P = 19 cm
Conclusion
Thus, the perimeter of triangle ABC is 19 cm, which corresponds to option 'a'.
In triangle ABC, we are given three equations involving the sides:
- AB + BC = 10 cm
- BC + CA = 12 cm
- CA + AB = 16 cm
Let’s denote the sides as follows:
- AB = a
- BC = b
- CA = c
Now, we can rewrite the equations based on these variables:
1. a + b = 10 (Equation 1)
2. b + c = 12 (Equation 2)
3. c + a = 16 (Equation 3)
Finding the Sides
To find the individual lengths of the sides, we can manipulate these equations:
- From Equation 1: a = 10 - b
- Substitute a in Equation 3: c + (10 - b) = 16
- This simplifies to c - b = 6 (Equation 4)
Next, we express c from Equation 2:
- From Equation 2: c = 12 - b (Equation 5)
Now we can substitute Equation 5 into Equation 4:
- (12 - b) - b = 6
- 12 - 2b = 6
- 2b = 6
- b = 3 cm
Calculating the Other Sides
Now that we have b, we can find a and c:
- Substitute b back into Equation 1:
- a + 3 = 10 → a = 7 cm
- Substitute b into Equation 5:
- c = 12 - 3 = 9 cm
Calculating the Perimeter
Now, we can find the perimeter P of triangle ABC:
- P = a + b + c
- P = 7 cm + 3 cm + 9 cm
- P = 19 cm
Conclusion
Thus, the perimeter of triangle ABC is 19 cm, which corresponds to option 'a'.
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In a ΔABC, if AB + BC = 10 cm, BC + CA = 12 cm, CA + AB = 16 cm, then the perimeter of the triangleis :-a)19 cmb)17 cmc)38 cmd)None of theseCorrect answer is option 'A'. Can you explain this answer?
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