SAT Math - Plane Geometry

Two chords of a circle, $\overline{AB}$ and $\overline{CD}$ , intersect at a point . $\overline{CX}$ is twice as long as $\overline{AX}$ , , and .

Give the length of $\overline{AX}$ .

Insufficient information is given to find the length of $\overline{AX}$ .

$10 \sqrt{2}$

$10 \sqrt{3}$

$20\sqrt{2}$

Explanation

Let stand for the length of $\overline{AX}$ ; then the length of $\overline{CX}$ is twice this, or . The figure referenced is below:

Chords

If two chords intersect inside the circle, then they cut each other in such a way that the product of the lengths of the parts is the same for the two chords - that is,

$AX \cdot BX = CX \cdot DX$

Substituting the appropriate quantities, then solving for :

$t \cdot 20 = 2t \cdot 10$

This statement is identically true. Therefore, without further information, we cannot determine the value of - the length of $\overline{AX}$ .

The length of the diagonal of a given square is $4\sqrt{2}$ . What is the square's area?

Explanation

If we divide the square into two triangles via its diagonal, then we know that the length of the diagonal is equal to the length of the triangles' hypotenuse.

We can use the Pythagorean Theorem to find the length of the two sides of one of our triangles.

Since we're dealing with a square, we know that the two sides of the square (which are the same as the two sides of one of our triangles) will be equal to one another. Therefore, we can say:

$(4\sqrt{2})^2 = 2a^2$

Now, solve for the unknown:

This means that the length of the sides of our triangle, as well as the sides of our square, is .

To find the area of the square, do the following:

$A = l\cdot w = 4 \cdot 4 = 16$ .

A triangle has the following side lengths:

$13,\ 13, \textup{ and } 18$

Which of the following correctly describes the triangle?

Acute and isosceles

Obtuse and isosceles

Acute and scalene

Obtuse and scalene

None of these

Explanation

The triangle has two sides of equal length, 13, so it is by definition isosceles.

To determine whether the triangle is acute, right, or obtuse, compare the sum of the squares of the lengths of the two shortest sides to the square of the length of the longest side. The former quantity is equal to

$13^{2} + 13^{2} = 169 + 169 = 338$

The latter quantity is equal to

$18^{2} = 324$

The former is greater than the latter; consequently, the triangle is acute. The correct response is that the triangle is acute and isosceles.

The base angle of an isosceles triangle is 15 less than three times the vertex angle. What is the vertex angle?

Explanation

Every triangle contains 180 degrees. An isosceles triangle has one vertex angle and two congruent base angles.

Let = vertex angle and = base angle

So the equation to solve becomes .

What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?

100

100√2

50√2

200√2

Explanation

Square_part1

Square_part2

Square_part3

The length of the diagonal of a given square is $4\sqrt{2}$ . What is the square's area?

Explanation

If we divide the square into two triangles via its diagonal, then we know that the length of the diagonal is equal to the length of the triangles' hypotenuse.

We can use the Pythagorean Theorem to find the length of the two sides of one of our triangles.

Since we're dealing with a square, we know that the two sides of the square (which are the same as the two sides of one of our triangles) will be equal to one another. Therefore, we can say:

$(4\sqrt{2})^2 = 2a^2$

Now, solve for the unknown:

This means that the length of the sides of our triangle, as well as the sides of our square, is .

To find the area of the square, do the following:

$A = l\cdot w = 4 \cdot 4 = 16$ .

A regular seven sided polygon has a side length of 14”. What is the measurement of one of the interior angles of the polygon?

128.57 degrees

257.14 degrees

180 degrees

154.28 degrees

252 degrees

Explanation

The formula for of interior angles based on a polygon with a number of side n is:

Each Interior Angle = (n-2)*180/n

= (7-2)*180/7 = 128.57 degrees

Solve each problem and decide which is the best of the choices given.

If $sin x =\frac{12}{13}$ , what is ?

$tan(x)=\frac{12}{5}$

$tan(x)=\frac{5}{13}$

$tan(x)=\frac{5}{12}$

$tan(x)=\frac{12}{13}$

$tan(x)=\frac{13}{12}$

Explanation

This is a triangle. We can find the value of the other leg by using the Pythagorean Theorem.

Remembering that

$sin=\frac{opposite}{hypotenuse}$ .

Thus,

If $sin(x)=\frac{12}{13}$ , you know the adjacent side is .

Thus, making

$tan(x)=\frac{opposite}{adjacent}=\frac{12}{5}$ because tangent is opposite/adjacent.

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

Act_math_01

8π - 16

4π-4

8π-4

2π-4

8π-8

Explanation

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

What is the hypotenuse of a right triangle with side lengths and ?