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In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is?
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In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a po...
Solution:
Given:
- Right triangle ABC right angled at B, AB = 1 and BC = 2.
- D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC.
To find:
- The area of triangle BDE as a fraction of the area of the triangle ABC.
Steps:
- Let's first draw a diagram to get a better understanding of the problem.
- From the diagram, we can see that triangle ABD and triangle BDE are similar as they have the same angles. So, we can say that:
BD/AB = DE/BD
BD^2 = AB * DE
BD^2 = 1 * DE
BD = DE^(1/2)
- Now, we can use the Pythagorean theorem to find the length of AC.
AC^2 = AB^2 + BC^2
AC^2 = 1^2 + 2^2
AC^2 = 5
AC = 5^(1/2)
- Using the area formula for triangle ABC, we get:
Area(ABC) = (1/2) * AB * BC
Area(ABC) = (1/2) * 1 * 2
Area(ABC) = 1
- Using the area formula for triangle BDE, we get:
Area(BDE) = (1/2) * BD * DE
Area(BDE) = (1/2) * DE^(1/2) * DE
Area(BDE) = (1/2) * DE^(3/2)
- To find the ratio of the areas, we divide the area of triangle BDE by the area of triangle ABC.
Ratio = Area(BDE)/Area(ABC)
Ratio = [(1/2) * DE^(3/2)]/1
Ratio = (1/2) * DE^(3/2)
- Substituting the value of DE, we get:
Ratio = (1/2) * (BD^2)^(3/4)
Ratio = (1/2) * BD^(3/2)
Ratio = (1/2) * (BD * BD^(1/2))
Ratio = (1/2) * (BD * DE^(1/2))
Ratio = (1/2) * BD * BD/BD
Ratio = 1/2
Answer:
- The area of triangle BDE as a fraction of the area of the triangle ABC is 1/2.
Given:
- Right triangle ABC right angled at B, AB = 1 and BC = 2.
- D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC.
To find:
- The area of triangle BDE as a fraction of the area of the triangle ABC.
Steps:
- Let's first draw a diagram to get a better understanding of the problem.
- From the diagram, we can see that triangle ABD and triangle BDE are similar as they have the same angles. So, we can say that:
BD/AB = DE/BD
BD^2 = AB * DE
BD^2 = 1 * DE
BD = DE^(1/2)
- Now, we can use the Pythagorean theorem to find the length of AC.
AC^2 = AB^2 + BC^2
AC^2 = 1^2 + 2^2
AC^2 = 5
AC = 5^(1/2)
- Using the area formula for triangle ABC, we get:
Area(ABC) = (1/2) * AB * BC
Area(ABC) = (1/2) * 1 * 2
Area(ABC) = 1
- Using the area formula for triangle BDE, we get:
Area(BDE) = (1/2) * BD * DE
Area(BDE) = (1/2) * DE^(1/2) * DE
Area(BDE) = (1/2) * DE^(3/2)
- To find the ratio of the areas, we divide the area of triangle BDE by the area of triangle ABC.
Ratio = Area(BDE)/Area(ABC)
Ratio = [(1/2) * DE^(3/2)]/1
Ratio = (1/2) * DE^(3/2)
- Substituting the value of DE, we get:
Ratio = (1/2) * (BD^2)^(3/4)
Ratio = (1/2) * BD^(3/2)
Ratio = (1/2) * (BD * BD^(1/2))
Ratio = (1/2) * (BD * DE^(1/2))
Ratio = (1/2) * BD * BD/BD
Ratio = 1/2
Answer:
- The area of triangle BDE as a fraction of the area of the triangle ABC is 1/2.
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In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is?
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In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is?.
In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a right triangle ABC right angled at B, AB = 1 and BC = 2.D is a point on AC such that BD = AB and E is the foot of perpendicular from D on BC. The area of triangle BDE as a fraction of the area of the triangle ABC is?.
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