: To prove, we draw a median to the given triangle. We use the definition of median. Then we’ll have two triangles with a common vertex and bases of equal length. Find the area of one triangle and show it is equal to the other.
Complete Step-by-Step solution:
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Let ABC be a triangle.
Let AD be one of its medians.
∆ABD and ∆ADC have the vertex A in common.
Hence, the bases BD and DC are equal (as AD is the median).
Now, draw a line AE perpendicular to BC, AE ⊥ BC.
We know the area of a triangle with base b and height h is = 12×b×h
Now area of triangle ∆ABD = 12×base× altitude of ∆ABD
= 12×BD×AE
= 12×DC×AE --- (Since BD = DC)
But DC and AE are the base and altitude of ∆ACD respectively.
Area of ∆ACD = 12× base DC × altitude of ∆ACD
= 12×DC×AE
Hence, area of (∆ABD) = area of (∆ACD)
Hence the median of a triangle divides it into two triangles of equal areas.
Note – The key in such problems is to draw a figure and include a median in it. This makes the figure into two triangles with a common vertex and equal bases.
Finding the area of one triangle and using the condition of equal bases gives us the proof.
(In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side.)
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Verified answer
: To prove, we draw a median to the given triangle. We use the definition of median. Then we’ll have two triangles with a common vertex and bases of equal length. Find the area of one triangle and show it is equal to the other.
Complete Step-by-Step solution:
seo images
Let ABC be a triangle.
Let AD be one of its medians.
∆ABD and ∆ADC have the vertex A in common.
Hence, the bases BD and DC are equal (as AD is the median).
Now, draw a line AE perpendicular to BC, AE ⊥ BC.
We know the area of a triangle with base b and height h is = 12×b×h
Now area of triangle ∆ABD = 12×base× altitude of ∆ABD
= 12×BD×AE
= 12×DC×AE --- (Since BD = DC)
But DC and AE are the base and altitude of ∆ACD respectively.
Area of ∆ACD = 12× base DC × altitude of ∆ACD
= 12×DC×AE
Hence, area of (∆ABD) = area of (∆ACD)
Hence the median of a triangle divides it into two triangles of equal areas.
Note – The key in such problems is to draw a figure and include a median in it. This makes the figure into two triangles with a common vertex and equal bases.
Finding the area of one triangle and using the condition of equal bases gives us the proof.
(In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side.)