In ABC, D and E are midpoints of side AB and AC respectively.Show that ar (ADE) =1/4 ar ( ABC)
Answers
Step-by-step explanation:
Draw line BE.
Consider triangles ABC and ABE. Since A, E, C lie in the same line, these triangles have the same perpendicular height from B to the base (AE or AC). Since the areas are half x base x height, the ratio between their areas is equal to the ratio between their bases. That is
area(ABC) / area(ABE) = AC / AE = 2, since E is the midpoint.
Similarly, by considering the triangle ABE and ADE which have the same perpendicular height from E, we have
area(ABE) / area(ADE) = AB / AD = 2, since D is the midpoint.
So
area(ABC) / area(ADE)
= ( area(ABC) / area(ABE) ) x ( area(ABE) / area(ADE) )
= 2 x 2 = 4.
That is, area(ABC) is 4 times greater than area(ADE).
Or in other words, area(ADE) = (1/4) area(ABC)
Answer:
proved
Step-by-step explanation:
Draw line BE.
Consider triangles ABC and ABE. Since A, E, C lie in the same line, these triangles have the same perpendicular height from B to the base (AE or AC). Since the areas are half x base x height, the ratio between their areas is equal to the ratio between their bases. That is
area(ABC) / area(ABE) = AC / AE = 2, since E is the midpoint.
Similarly, by considering the triangle ABE and ADE which have the same perpendicular height from E, we have
area(ABE) / area(ADE) = AB / AD = 2, since D is the midpoint.
So
area(ABC) / area(ADE)
= ( area(ABC) / area(ABE) ) x ( area(ABE) / area(ADE) )
= 2 x 2 = 4.
That is, area(ABC) is 4 times greater than area(ADE).
Or in other words, area(ADE) = (1/4) area(ABC)
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