ACT Math : Quadrilaterals
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Example Question #231 : Quadrilaterals
A quadrilateral ABCD has diagonals that are perpendicular bisectors of one another. Which of the following classifications must apply to quadrilateral ABCD?
I. parallelogram
II. rhombus
III. square
I, II, and III
II and III only
I and III only
I and II only
I and II only
If the diagonals of a quadrilateral are perpendicular bisectors of one another, then the quadrilateral must be a rhombus, but not necessarily a square. Since all rhombi are also parallelograms, quadrilateral ABCD must be both a rhombus and parallelogram.
Example Question #32 : Rhombuses
Are the two rhombuses in the above picture similar?
Maybe
Not enough information to decide.
No
Yes
No
Two shapes are similar to each other if they are the same except for differences in scaling. This means that all of the angles of one shape have to be the same as the angles of the other shape, and all the sides have to be proportional to each other. For example, if we have two rectangles A and B, where A has side lengths 2 and 4, and B has side lengths 5 and 10, A and B are similar because the ratio between 2 and 4 is the same as the ratio between 5 and 10, and also the ratio between 2 and 5 is the same as the ratio between 4 and 10. Everything is proportional (and all the angles are the same) so the two rectangles A and B are similar.
The two rhombuses in our figure are not similar to each other, because we can see the larger angle in the rhombus on the left is not the same as the larger angle in the rhombus on the right. If we scaled the larger rhombus down and tried to cover up the smaller rhombus with it, it wouldn't work because they aren't proportional to each other.
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