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The area of the quadrilateral ABCD whose vertices are respectively A(1,1), B(7,-3), C(12,2) and D(7,21) is … Sq. units
- a)108
- b)127
- c)132
- d)144
Correct answer is option 'C'. Can you explain this answer?
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The area of the quadrilateral ABCD whose vertices are respectively A(1...
To find the area of the quadrilateral ABCD, we can divide it into two triangles and find the sum of their areas.
Triangle ABC:
The length of AB can be found using the distance formula:
AB = sqrt((7-1)^2 + (-3-1)^2) = sqrt(6^2 + 4^2) = sqrt(36 + 16) = sqrt(52)
The length of AC can also be found using the distance formula:
AC = sqrt((12-1)^2 + (2-1)^2) = sqrt(11^2 + 1^2) = sqrt(121 + 1) = sqrt(122)
The angle between AB and AC can be found using the dot product formula:
cos(θ) = (AB · AC) / (|AB| |AC|)
cos(θ) = ((7-1)(12-1) + (-3-1)(2-1)) / (sqrt(52) sqrt(122))
cos(θ) = (6*11 + (-4)*1) / (sqrt(52) sqrt(122))
cos(θ) = (66 - 4) / (sqrt(52) sqrt(122))
cos(θ) = 62 / (sqrt(52) sqrt(122))
The area of triangle ABC is given by the formula:
Area_ABC = 0.5 |AB| |AC| sin(θ)
Area_ABC = 0.5 sqrt(52) sqrt(122) sin(θ)
Area_ABC = 0.5 sqrt(52) sqrt(122) sqrt(1 - cos^2(θ))
Area_ABC = 0.5 sqrt(52) sqrt(122) sqrt(1 - (62 / (sqrt(52) sqrt(122)))^2)
Triangle ACD:
The length of AD can be found using the distance formula:
AD = sqrt((7-1)^2 + (21-1)^2) = sqrt(6^2 + 20^2) = sqrt(36 + 400) = sqrt(436)
The length of AC remains the same as before.
The angle between AD and AC can be found using the dot product formula:
cos(θ) = (AD · AC) / (|AD| |AC|)
cos(θ) = ((7-1)(12-1) + (21-1)(2-1)) / (sqrt(436) sqrt(122))
cos(θ) = (6*11 + 20*1) / (sqrt(436) sqrt(122))
cos(θ) = (66 + 20) / (sqrt(436) sqrt(122))
cos(θ) = 86 / (sqrt(436) sqrt(122))
The area of triangle ACD is given by the formula:
Area_ACD = 0.5 |AD| |AC| sin(θ)
Area_ACD = 0.5 sqrt(436) sqrt(122) sin(θ)
Area_ACD = 0.5 sqrt(436) sqrt(122) sqrt(1 - cos^2(θ))
Area_ACD = 0.5 sqrt(436) sqrt(122) sqrt(1 - (86 / (sqrt(436) sqrt(122)))^2)
The area of the quadrilateral ABCD is the sum of the areas of triangles ABC
Triangle ABC:
The length of AB can be found using the distance formula:
AB = sqrt((7-1)^2 + (-3-1)^2) = sqrt(6^2 + 4^2) = sqrt(36 + 16) = sqrt(52)
The length of AC can also be found using the distance formula:
AC = sqrt((12-1)^2 + (2-1)^2) = sqrt(11^2 + 1^2) = sqrt(121 + 1) = sqrt(122)
The angle between AB and AC can be found using the dot product formula:
cos(θ) = (AB · AC) / (|AB| |AC|)
cos(θ) = ((7-1)(12-1) + (-3-1)(2-1)) / (sqrt(52) sqrt(122))
cos(θ) = (6*11 + (-4)*1) / (sqrt(52) sqrt(122))
cos(θ) = (66 - 4) / (sqrt(52) sqrt(122))
cos(θ) = 62 / (sqrt(52) sqrt(122))
The area of triangle ABC is given by the formula:
Area_ABC = 0.5 |AB| |AC| sin(θ)
Area_ABC = 0.5 sqrt(52) sqrt(122) sin(θ)
Area_ABC = 0.5 sqrt(52) sqrt(122) sqrt(1 - cos^2(θ))
Area_ABC = 0.5 sqrt(52) sqrt(122) sqrt(1 - (62 / (sqrt(52) sqrt(122)))^2)
Triangle ACD:
The length of AD can be found using the distance formula:
AD = sqrt((7-1)^2 + (21-1)^2) = sqrt(6^2 + 20^2) = sqrt(36 + 400) = sqrt(436)
The length of AC remains the same as before.
The angle between AD and AC can be found using the dot product formula:
cos(θ) = (AD · AC) / (|AD| |AC|)
cos(θ) = ((7-1)(12-1) + (21-1)(2-1)) / (sqrt(436) sqrt(122))
cos(θ) = (6*11 + 20*1) / (sqrt(436) sqrt(122))
cos(θ) = (66 + 20) / (sqrt(436) sqrt(122))
cos(θ) = 86 / (sqrt(436) sqrt(122))
The area of triangle ACD is given by the formula:
Area_ACD = 0.5 |AD| |AC| sin(θ)
Area_ACD = 0.5 sqrt(436) sqrt(122) sin(θ)
Area_ACD = 0.5 sqrt(436) sqrt(122) sqrt(1 - cos^2(θ))
Area_ACD = 0.5 sqrt(436) sqrt(122) sqrt(1 - (86 / (sqrt(436) sqrt(122)))^2)
The area of the quadrilateral ABCD is the sum of the areas of triangles ABC
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The area of the quadrilateral ABCD whose vertices are respectively A(1,1), B(7,-3), C(12,2) and D(7,21) is … Sq. unitsa)108b)127c)132d)144Correct answer is option 'C'. Can you explain this answer?
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