SAT Math - 2-Dimensional Geometry

Find the area of a circle with a diameter of .

$64\pi$

$32\pi$

$16\pi$

$81\pi$

$256\pi$

Explanation

Write the formula for the area of a circle.

$A_{circle}= \pi r^2 = \frac{\pi d^2}{4}$

Substitute the diameter and solve.

$\frac{\pi (16)^2}{4} = \frac{\pi (16)(16)}{4}=\pi (16)(4)= 64\pi$

A circle has a diameter of 10cm. What is the circumference?

$10\pi cm$

$20\pi cm$

$5\pi cm$

$25\pi cm$

$100\pi cm$

Explanation

The circumference of a circle is given by the equation:

$C=2\pi r$

The radius is half the diameter, in this case half of 10cm is 5cm

Plug in 5cm for r

$C=2\pi 5cm$

Simplify to get the final answer

$C=10\pi cm$

On the XY plane, line segment AB has endpoints (0, a) and (b, 0). If a square is drawn with segment AB as a side, in terms of a and b what is the area of the square?

$A=\frac{1}{2} (a + b)$

Cannot be determined

Explanation

Since the question is asking for area of the square with side length AB, recall the formula for the area of a square.

$A=\text{side}^2$

Now, use the distance formula to calculate the length of AB.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

let

$\(x_1,y_1)=(0,a) \(x_2,y_2)=(b,0)$

Now substitute the values into the distance formula.

$\d=\sqrt{(b-0)^2+(0-2)^2} \d=\sqrt{b^2+a^2}$

From here substitute the side length value into the area formula.

$\A=\text{side}^2 \A=\left(\sqrt{b^2+a^2}\right)^2 \A=b^2+a^2$

Find the area of a kite with diagonal lengths of and .

Explanation

Write the formula for the area of a kite.

$A= \frac{d1\cdot d2}{2}$

Plug in the given diagonals.

$A= \frac{(a+b)\cdot (2a-2b)}{2}$

Pull out a common factor of two in and simplify.

$A= \frac{(a+b)\cdot2(a-b)}{2}$

Use the FOIL method to simplify.

Which of the following describes a triangle with sides one kilometer, 100 meters, and 100 meters?

The triangle cannot exist.

The triangle is acute and equilateral.

The triangle is obtuse and isosceles, but not equilateral.

The triangle is acute and isosceles, but not equilateral.

The triangle is obtuse and scalene.

Explanation

One kilometer is equal to 1,000 meters, so the triangle has sides of length 100, 100, and 1,000. However,

That is, the sum of the least two sidelengths is not greater than the third. This violates the Triangle Inequality, and this triangle cannot exist.

Triangle

Note: Figure NOT drawn to scale.

Refer to the figure above, which shows a square inscribed inside a large triangle. What percent of the entire triangle has been shaded blue?

$44\frac{4}{9} %$

$33 \frac{1}{3} %$

Insufficient information is given to answer the question.

Explanation

The shaded portion of the entire triangle is similar to the entire large triangle by the Angle-Angle postulate, so sides are in proportion. The short leg of the blue triangle has length 20; that of the large triangle, 30. Therefore, the similarity ratio is . The ratio of the areas is the square of this, or $3^{2}:2^{2}$ , or .