Question 1160335: The areas of two similar triangles are 108 cm² and 972 cm². What is the ratio of the lengths of their corresponding sides?
Found 3 solutions by greenestamps, ikleyn, Theo:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The ratio of the areas is 108:972 = 1:9.
By a powerful feature of similar figures, the ratio of corresponding linear measurements is sqrt(1):sqrt(9) = 1:3.
ANSWER: 1:3
Answer by ikleyn(52781)  (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! if tbhe triangles are similar, then their corresponding sides are proportional.
that means that the corresponding sides of one of the triangle is equal to x * the corresponding side of the other triangle.
the area of a triangle is equal to 1/2 * b * h
b is one of the side.
h is the other of the sides.
since the triangles are similar, then the sides of the second triangle would be x * the corresponding side.
therefore, the area of the first triangle is 1/2 * b * h and the area of the second triangle is 1/2 * x * b * x * h.
this simplifies to 1/2 * x^2 * b * h.
shown another way, it is equal to 1/2 * b * h * x^2.
if the area of the first triangle is 108 and the area of the second triangle is 972, then x^2 = 972 / 108 = 9
this makes x = 3.
the ratio of their corresponding sides is therefore 3.
to confirm, assume 1/2 * b * h = 108
let b = 12 and h = 18
you get 1/2 * b * h = 1/2 * 12 * 18 = 108.
since x = 3, then the dimensions for the second triangle are:
b = 3 * 12 = 36
h = 3 * 18 = 54
the area of the second triangle is equal to 1/2 * 36 * 54 = 972.
the ratio of the areas is 972 / 108 = 9
the ratios of the corresponding sides is 3 which is the square root of the ratio of the area.
your solution is that the ratio of the area of the corresponding sides is the square root of the ratio of the areas which is the square root of 9 which is 3.
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