SSAT Upper Level Quantitative - How to find the area of a parallelogram

Parallelogram

Figure NOT drawn to scale.

The above depicts Rhombus , which has perimeter 80. .

Give the area of Rhombus .

Explanation

The area of any parallelogram is the product of the length of its base and that of a corresponding altitude. We can take $\overline{CD}$ as a base and perpendicular $\overline{XC}$ as an altitude.

All four sides of a rhombus have the same length, so we can find by dividing the perimeter - the sum of the lengths of the four sides - by 4:

$CD = p \div 4 = 80 \div 4 = 20$

Now multiply the lengths of this base and the altitude to get the area:

$a = CD \cdot XC = 20 \cdot 16 = 320$

The length of the shorter diagonal of a rhombus is 40% that of the longer diagonal. The area of the rhombus is . Give the length of the longer diagonal in terms of .

$\sqrt{5K}$

$\sqrt{\frac{5K}{2}}$

$K\sqrt{\frac{5}{2}}$

$\frac{1}{5}\sqrt{K}$

$K\sqrt{\frac{2}{5}}$

Explanation

Let be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to $\frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}$ , 40% of is equal to $\frac{2}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for , in the equation:

$\frac{1}{2} \cdot \frac{2}{5}D \cdot D = K$

$\frac{1}{5}D^{2}= K$

$5\cdot \frac{1}{5}D^{2}=5\cdot K$

$D^{2}=5K$

$D =\sqrt{5K}$

Which of the following shapes is NOT a quadrilateral?

Triangle

Square

Rectangle

Rhombus

Kite

Explanation

A quadrilateral is any two-dimensional shape with sides. The only shape listed that does not have sides is a triangle.

Which shape is NOT a quadrilateral?

Circle

Trapezoid

Kite

Rhombus

Rectangle

Explanation

A quadrilateral has to have sides, a circle does not have any sides.

Which two shapes have to have right angles?

Square and Rectangle

Square and Rhombus

Square and Parallelogram

Rectangle and Rhombus

Rectangle and Parallelogram

Explanation

By definition, the only two quadrilaterals that have to have right angles, are the square and the rectangle.

What is the difference between a trapezoid and a isosceles trapezoid?

An isosceles trapezoid has to have equal base angles

A trapezoid has to have equal base angles

An isosceles trapezoid has to have equal side lengths

A trapezoid has to have equal side lengths

There is no difference between them

Explanation

By definition, an isosceles trapezoid has to have equal base angles, but a trapezoid does not have to have equal angles.

A parallelogram has the base length of and the altitude of . Give the area of the parallelogram.

Explanation

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. So we can write:

A parallelogram has a base length of which is 3 times longer than its corresponding altitude. The area of the parallelogram is 12 square inches. Give the .

$6\sqrt{2}$

$6\sqrt{3}$

Explanation

Base length is so the corresponding altitude is $\frac{t}{3}$ .

The area of a parallelogram is given by:

Where:

is the length of any base
is the corresponding altitude

So we can write:

$12=t\times \frac{t}{3}\Rightarrow t\times t=12\times 3\Rightarrow t^2=36\Rightarrow t=6$

$t\times t=12\times 3$

$t^{2}=36$

The longer diagonal of a rhombus is 20% longer than the shorter diagonal; the rhombus has area . Give the length of the shorter diagonal in terms of .

$\sqrt{ \frac{5N}{3}}$

$\sqrt{ \frac{3N}{5}}$

$\sqrt{ \frac{N}{5}}$

$\sqrt{5N}$

$\sqrt{15N}$

Explanation

Let be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is

$\frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}$

of , or $\frac{6}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for :

$\frac{1}{2} \cdot \frac{6}{5}D \cdot D = N$

$\frac{3}{5}D^{2} = N$

$\frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N$

$D^{2} = \frac{5N}{3}$

$D =\sqrt{ \frac{5N}{3}}$

The length of the shorter diagonal of a rhombus is two-thirds that of the longer diagonal. The area of the rhombus is square yards. Give the length of the longer diagonal, in inches, in terms of .