2 Questions around this concept.
If triangles ABC and DBC are on the same base BC and AD intersects BC at O
then
In the given figure, the line segment XY is parallel to the side AC of the triangle ABC and it divides the triangle into two parts of equal area. Then AX:AB is
You have learnt that in two similar triangles, the ratio of their corresponding sides is the same. Do you think there is any relationship between the ratio of their areas and the ratio of the corresponding sides?
You know that area is measured in square units. So, you may expect that this ratio is the square of the ratio of their corresponding sides. This is indeed true and we shall prove it in the next theorem.
Thoerem : The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Given:
To prove:
Construction:
Prrof:
Form (i) and (ii)
Important Results on Area of Similar Triangle:
1. The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes.
In the following figures, ΔABC ~ ΔDEF and AX, DY are the altitudes.
2. The ratio of areas of two similar triangles is equal to the ratio of the squares on their corresponding medians.
In the following figure, ΔABC ~ ΔPQR and AD and PS are medians.
3. The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisector segments.
In the figure, ΔABC ~ ΔDEF and AP, DQ are bisectors of ∠A and ∠D respectively,
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