Geometry - Intermediate Geometry

If the -intercept of a line is , and the -intercept is , what is the equation of this line?

$y=-\frac{1}{5}x+5$

$y=\frac{1}{5}x+5$

Explanation

If the y-intercept of a line is , then the -value is when is zero. Write the point:

If the -intercept of a line is , then the -value is when is zero. Write the point:

Use the following formula for slope and the two points to determine the slope:

$m=\frac{y_2-y_1}{x_2-x_1} = \frac{0-5}{-1-0}= \frac{-5}{-1} =5$

Use the slope intercept form and one of the points, suppose , to find the equation of the line by substituting in the values of the point and solving for , the -intercept.

Therefore, the equation of this line is .

Each side of this pentagon has a length of .

Solve for the area of the pentagon.

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Explanation

The formula for area of a pentagon is $A=5\cdot(\frac{1}{2}\cdot s\cdot h)$ , with representing the length of one side and representing the apothem.

To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:

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Each of these triangles have angle measures of $72$^{\circ}$; 54$^{\circ}$;54$^{\circ}$$ , with $72$^{\circ}$$ being the angle oriented around the vertex. This is because the polygon has been divided into five triangles and $\frac{360}{5}= 72$ .

Varsity pentagon

To solve for the apothem, we can use basic trigonometric ratios:

$tan(54$^{\circ}$)= 1.3764$

$\1.3764=\frac{x}{2.5;cm}\x=3.44;cm$

Now that we know the apothem length, we can plug in all our values to solve for area:

$Area=5\cdot\frac{1}{2}\cdot 5;cm \cdot3.44;cm=43;cm$

If a triangle has side lengths of and , which of the following can be a length of the third side?

Explanation

The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. The relationship can be represented by the following inequalities:

The side length of is the only choice that fits this criteria:

A circle graph is used to depict the results from a survey given to high school students regarding their favorite sport. The sector representing baseball measures 52 degrees. What percent of the high schoolers who took the survey prefer baseball over any other sport?

None of the other answers.

Explanation

To convert the angle of a sector in a circle to the percentage of that sector you can set-up a proportion: $\frac{52}{360} = \frac{x}{100}$ and then solve. The total number of degrees in a circle is 360, so you place 52 out of 360. The total percent being represented by a circle is 100, so you place (what you are trying to find) out of 100.

Rhombus_1

The above figure shows a rhombus . Give its area.

$40 \sqrt{11}$

$40\sqrt{61}$

Explanation

Construct the other diagonal of the rhombus, which, along with the first one, form a pair of mutual perpendicular bisectors.

Rhombus_1

By the Pythagorean Theorem,

$n = \sqrt{12^{2}- 10^{2}} =\sqrt{144- 100} = \sqrt{44}= \sqrt{4}\cdot \sqrt{11} = 2 \sqrt{11}$

The rhombus can be seen as the composite of four congruent right triangles, each with legs 10 and $2 \sqrt{11}$ , so the area of the rhombus is

$A = 4 \cdot \frac{1}{2}\cdot 10 \cdot 2\sqrt{11} =40 \sqrt{11}$ .

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

True

False

Explanation

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

A triangle is equilateral (having three sides of the same length) if and only if it is also equiangular (having three angles of the same measure, each of which is $60^{\circ }$ ). It follows that all angles of both triangles measure $60^{\circ }$ .

Specifically, $\angle A \cong \angle D$ and $\angle B \cong \angle E$ , making two pairs of corresponding angles congruent. By the Angle-Angle Similarity Postulate, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ , making the correct answer "true".

A sphere with a radius of is cut out of a cube that has a side edge of . What is the volume of the resulting shape?

Explanation

Since the cube is bigger, we will be subtracting the volume of the sphere from the volume of the cube.

Start by recalling how to find the volume of a sphere.

$\text{Volume of Sphere}=\frac{4}{3}\pi r^3$

Plug in the given radius to find the volume.

$\text{Volume of Sphere}=\frac{4}{3}\pi(2)^3=\frac{32}{3}\pi$

Next, recall how to find the volume of a cube:

$\text{Volume of Cube}=(\text{edge})^3$

Plug in the given side length to find the volume of the cube.

$\text{Volume of Cube}=4^3=64$

Finally, subtract the volume of the sphere from the volume of the cube.

$\text{Volume of Figure}=64-\frac{32}{3}\pi=30.49$

Make sure to round to places after the decimal.

$\bigtriangleup ABC$ is an equilateral triangle; $\bigtriangleup DEF$ is an equiangular triangle.

True or false: From the given information, it follows that $\bigtriangleup ABC \sim \bigtriangleup DEF$ .

True

False

Explanation

As we are establishing whether or not $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then , , and correspond respectively to , , and .

If the height of the parallelogram is half of the length of the rectangle, then find the area of the shaded region in the figure.

Explanation

In order to find the area of the shaded region, we must first find the areas of the rectangle and parallelogram.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$

Substitute in the given length and height to find the area of the rectangle.

$\text{Area of Rectangle}=8 \times 14$

$\text{Area of Rectangle}=112$

Next, find the area of the parallelogram.

Recall how to find the area of a parallelogram:

$\text{Area of Parallelogram}=\text{base}\times\text{height}$

We need to find the height of the parallelogram. From the question, we know the following relationship:

$\text{height}=\frac{\text{length}}{2}$

Substitute in the length of the rectangle to find the height of the parallelogram.

$\text{height}=\frac{8}{2}$

$\text{height}=4$

Now, substitute in the height and the given length of the base to find the area of the parallelogram.

$\text{Area of Parallelogram}=10 \times 4$

$\text{Area of Parallelogram}=40$

Now, we are ready to find the area of the shaded region by subtracting the area of the parallelogram from the area of the rectangle.

$\text{Area of Shaded Region}=\text{Area of Rectangle}-\text{Area of Parallelogram}$

$\text{Area of Shaded Region}=112-40$

Which of the following points is found on the line ?

Explanation

To find which point is on the line, take the -coordinate, and plug it into the given equation to solve for . If the -value matches the -coordinate of the same point, then the point is on the line.