Quadrilaterals
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ISEE Upper Level Quantitative Reasoning › Quadrilaterals
A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?
decreased by 4%
increased by 20%
the area remains the same
decreased by 10%
Explanation
The area decreases by 20% of 20%, which is 4%.
The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.
10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.
The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.
Find the area of the above trapezoid if 
Figure not drawn to scale.
Explanation
The area of a trapezoid is given by
where 
Three of the vertices of a parallelogram on the coordinate plane are 
Insufficient information is given to answer the problem.
Explanation
As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:
Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.
The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:
The area of the 12 by 12 square is
The area of the green triangle is 
The area of the blue triangle is 
The area of the pink triangle is 
The area of the main triangle is therefore
The parallelogram has area twice this, or 
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?
Explanation
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?
Let's begin with our formula for the area of a square:
where s is our side length and A is our area.
With this formula, we can solve for our side length by plugging in our area and square rooting both sides.
Three of the vertices of a parallelogram on the coordinate plane are
Insufficient information is given to answer the problem.
Explanation
As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:
Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.
The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:
The area of the 12 by 12 square is
The area of the green triangle is
The area of the blue triangle is
The area of the pink triangle is
The area of the main triangle is therefore
The parallelogram has area twice this, or
Find the area of the above trapezoid if
Figure not drawn to scale.
Explanation
The area of a trapezoid is given by
where
A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?
decreased by 4%
increased by 20%
the area remains the same
decreased by 10%
Explanation
The area decreases by 20% of 20%, which is 4%.
The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.
10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.
The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.
Find the area of the above trapezoid if
Figure not drawn to scale.
Explanation
The area of a trapezoid is given by
where
Three of the vertices of a parallelogram on the coordinate plane are
Insufficient information is given to answer the problem.
Explanation
As can be seen in the diagram, there are three possible locations of the fourth point of the parallelogram:
Regardless of the location of the fourth point, however, the triangle with the given three vertices comprises exactly half the parallelogram. Therefore, the parallelogram has double that of the triangle.
The area of the triangle can be computed by noting that the triangle is actually a part of a 12-by-12 square with three additional right triangles cut out:
The area of the 12 by 12 square is
The area of the green triangle is
The area of the blue triangle is
The area of the pink triangle is
The area of the main triangle is therefore
The parallelogram has area twice this, or
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?
Explanation
Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the length of one side of her room?
Let's begin with our formula for the area of a square:
where s is our side length and A is our area.
With this formula, we can solve for our side length by plugging in our area and square rooting both sides.