ISEE Upper Level Quantitative : Geometry
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Example Questions
Example Question #1 : How To Find The Length Of The Diagonal Of A Rhombus
A rhombus has sidelength ten inches; one of its diagonals is one foot long. Which is the greater quantity?
(a) The length of the other diagonal
(B) One and one-half feet
(A) and (B) are equal
It is impossible to determine which is greater from the information given
(B) is greater
(A) is greater
(B) is greater
The diagonals of a rhombus are each other's perpendicular bisector, so, as can be seen in the diagram below, one side of a rhombus and one half of each diagonal form a right triangle. If the other diagonal has length 
This triangle fits the well-known Pythagorean triple of 6-8-10, so
The other diagonal measures 16 inches. One and one-half feet is equal to 18 inches, making (B) greater.
Example Question #63 : Quadrilaterals
Rhombus 
(a)
(b)
(b) is greater
It is impossible to tell from the information given
(a) is greater
(a) and (b) are equal
(a) and (b) are equal
The diagonals of a rhombus always intersect at right angles, so 
Example Question #1 : How To Find The Area Of A Rectangle
Which is the greater quantity?
(a) The area of the rectangle on the coordinate plane with vertices
(b) The area of the rectangle on the coordinate plane with vertices
It is impossible to tell from the information given.
(b) is greater.
(a) is greater.
(a) and (b) are equal.
(a) and (b) are equal.
(a) The first rectangle has width 
(b) The second rectangle has width 
The areas of the rectangle are the same.
Example Question #1 : How To Find The Area Of A Rectangle
A rectangle on the coordinate plane has its vertices at the points 
Which is the greater quantity?
(a) The area of the portion of the rectangle in Quadrant I
(b) The area of the portion of the rectangle in Quadrant III
It is impossible to tell from the information given.
(a) and (b) are equal.
(a) is greater.
(b) is greater.
(a) and (b) are equal.
(a) The portion of the rectangle in Quadrant I is a rectangle with vertices 
(a) The portion of the rectangle in Quadrant III is a rectangle with vertices 
Example Question #2 : How To Find The Area Of A Rectangle
(a) The area of the square on the coordinate plane with vertices
(b) The area of the rectangle on the coordinate plane with vertices
(b) is greater.
(a) and (b) are equal.
It is impossible to tell from the information given.
(a) is greater.
(a) is greater.
(a) The square has sidelength 
(b) The rectangle has width 
Since 
Example Question #1 : Rectangles
The perimeter of a rectangle is one yard. The rectangle is three times as long as it is wide. Which is the greater quantity?
(a) The area of the rectangle
(b) 60 square inches
(b) is greater
(a) is greater
(a) and (b) are equal
It is impossible to tell form the information given
(a) is greater
Let 
Since the perimeter is one yard, or 36 inches,
Example Question #3 : How To Find The Area Of A Rectangle
If one rectangular park measures 
First, you must compute the area of both parks. The area of a rectangle is length times width. Therefore, the area of park one is 
Example Question #6 : How To Find The Area Of A Rectangle
The sum of the lengths of three sides of a rectangle is 572 inches; the width of the rectangle is 60% of its length. Give its area in square inches.
It is impossible to determine the area from the given information.
It is impossible to determine the area from the given information.
Since the width of the rectangle is 60% of its length, we can write 
However, it is not clear from the problem which three sides - two lengths and a width or two widths and a length - we are choosing to have sum 572 inches. Depending on the three sides chosen, we can either set up
or
Since the length cannot be determined with certainty, neither can the width, and, subsequently, neither can the area.
Example Question #7 : How To Find The Area Of A Rectangle
Five rectangles each have the same length, which we will call 
The area of a rectangle is the product of its width and its length. The areas of the five rectangles, therefore, are 
Example Question #8 : How To Find The Area Of A Rectangle
Two rectangles, A and B, each have perimeter 32 feet. Rectangle A has length 12 feet; Rectangle B has length 10 feet. The area of Rectangle A is what percent of the area of Rectangle B?
The perimeter of a rectangle can be given by the formula
Since for both rectangles, 30 is the perimeter, this becomes
Rectangle A has length 12 feet and, subsequently. width 4 feet, making its area
Rectangle B has length 10 feet and, subsequently. width 6 feet, making its area
The area of Rectangle A is
of the area of Rectangle B.
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