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In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is?
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In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points ...
In order to find the perimeter of triangle ELD, we first need to determine the lengths of the sides EL, LD, and DE.
1. Finding the length of EL:
Since DN is the perpendicular bisector of AC, it divides AC into two congruent segments. Therefore, AD = DC. Similarly, since EM is the perpendicular bisector of AB, it divides AB into two congruent segments. Therefore, AE = EB.
Since ABC is a triangle with BC = 24 and angle A = 60 degrees, we can use the Law of Sines to find the length of AB.
sin A / AB = sin B / BC
sin 60° / AB = sin B / 24
√3 / AB = sin B / 24
AB = (24 * √3) / sin B
Since AE = EB, we have EL = (AB - AE) / 2 = (AB - AB) / 2 = 0.
2. Finding the length of LD:
Since DN is the perpendicular bisector of AC, DN is also the altitude of triangle ADC. Therefore, triangle ADC is a right-angled triangle with angle A = 60 degrees and angle D = 90 degrees. Thus, triangle ADC is a 30-60-90 triangle, and we can use the ratios of the sides to find the length of AD and DC.
Let AD = x, then DC = x.
In a 30-60-90 triangle, the ratio of the sides is x : x√3 : 2x.
Therefore, AD = DC = x = 24 / 2 = 12.
Since L is the midpoint of MN, we have LD = MN / 2.
3. Finding the length of DE:
Since EM is the perpendicular bisector of AB, EM is also the altitude of triangle ADE. Therefore, triangle ADE is a right-angled triangle with angle A = 60 degrees and angle E = 90 degrees. Thus, triangle ADE is a 30-60-90 triangle, and we can use the ratios of the sides to find the length of AE and DE.
Let AE = y, then DE = y√3.
In a 30-60-90 triangle, the ratio of the sides is y : y√3 : 2y.
Therefore, AE = y = 24 / 2√3 = 4√3.
Since L is the midpoint of MN, we have LE = MN / 2.
4. Finding the perimeter of triangle ELD:
The perimeter of triangle ELD is EL + LD + DE.
Since EL = 0, we only need to find the lengths of LD and DE.
LD = MN / 2 = 12 / 2 = 6.
DE = y√3 = 4√3√3 = 4 * 3 = 12.
Therefore, the perimeter of triangle ELD is 0 + 6 + 12 = 18 units.
1. Finding the length of EL:
Since DN is the perpendicular bisector of AC, it divides AC into two congruent segments. Therefore, AD = DC. Similarly, since EM is the perpendicular bisector of AB, it divides AB into two congruent segments. Therefore, AE = EB.
Since ABC is a triangle with BC = 24 and angle A = 60 degrees, we can use the Law of Sines to find the length of AB.
sin A / AB = sin B / BC
sin 60° / AB = sin B / 24
√3 / AB = sin B / 24
AB = (24 * √3) / sin B
Since AE = EB, we have EL = (AB - AE) / 2 = (AB - AB) / 2 = 0.
2. Finding the length of LD:
Since DN is the perpendicular bisector of AC, DN is also the altitude of triangle ADC. Therefore, triangle ADC is a right-angled triangle with angle A = 60 degrees and angle D = 90 degrees. Thus, triangle ADC is a 30-60-90 triangle, and we can use the ratios of the sides to find the length of AD and DC.
Let AD = x, then DC = x.
In a 30-60-90 triangle, the ratio of the sides is x : x√3 : 2x.
Therefore, AD = DC = x = 24 / 2 = 12.
Since L is the midpoint of MN, we have LD = MN / 2.
3. Finding the length of DE:
Since EM is the perpendicular bisector of AB, EM is also the altitude of triangle ADE. Therefore, triangle ADE is a right-angled triangle with angle A = 60 degrees and angle E = 90 degrees. Thus, triangle ADE is a 30-60-90 triangle, and we can use the ratios of the sides to find the length of AE and DE.
Let AE = y, then DE = y√3.
In a 30-60-90 triangle, the ratio of the sides is y : y√3 : 2y.
Therefore, AE = y = 24 / 2√3 = 4√3.
Since L is the midpoint of MN, we have LE = MN / 2.
4. Finding the perimeter of triangle ELD:
The perimeter of triangle ELD is EL + LD + DE.
Since EL = 0, we only need to find the lengths of LD and DE.
LD = MN / 2 = 12 / 2 = 6.
DE = y√3 = 4√3√3 = 4 * 3 = 12.
Therefore, the perimeter of triangle ELD is 0 + 6 + 12 = 18 units.
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In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is?
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In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is?.
In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is?.
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In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is?, a detailed solution for In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is? has been provided alongside types of In a triangle ABC , BC=24 and angle A=60 degree . D ,M are the points on side AC and E , N are the points on AB such that DN and EM are the perpendicular bisectors of AC and AB respectively . If L be the mid-point of MN , then the perimeter of triangle ELD is? theory, EduRev gives you an
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