Geometry - ACT Math

Based on the description of your triangle, you can draw the following figure:

You can do this because you know:

1. The two equivalent sides are given.
2. Since a triangle is degrees, you have only or degrees left for the two angles of equal size. Therefore, those two angles must be degrees and degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

Based on your standard reference triangle, you know:

Therefore, is .

This means that is , and the total base of the triangle is .

Now, the area of the triangle is:

or

To find the speed of the deer, you must have the distance traveled and the time.

The distance is found using the Pythagorean Theorem:

The answer must be in miles per hour so the total miles are divided by the hours to get the final answer:

To find the slope, put the line in slope intercept form. In other words put the equation in form where represents the slope and represents the y-intercept.

From here we can see our slope equals :

The expression used in solving this question is the distance formula:

This formula is simply a variation of the Pythagorian Theorem. A great way to remember this formula is to visualize a right triangle where two of the vertices are the points given in the problem statement. For this question:

Where a = and b = . Now it should be easy to see how the distance formula is simply a variation of the Pythagorean Theorem.

We almost have all of the information we need to solve the problem, but we still need to find the coordinates of the triangle at the right angle. This can be done by simply taking the y-coordinate of the first point and the x-coordinate of the second point, resulting in .

Now we can simply plug and chug using the distance formula.

In order to solve this problem, we need to find a point that will satisfy both equations. In order to do this, we need to combine the two equations into a single expression. For this, we need to isolate either x or y in one of the equations. Since the equation already has y isolated, we will use this equation. Next we substitue this equation into the first one. becomes which simplifies to . Now we can solve for x by factoring: Thus, .

Now that we have two possible values for x, we can plug each value into either equation to obtain two values for y. For and the second equation, we get . Therefore our first point is . This is not one of the listed answers, so we will use our other value of x. For and the second equation, we get . This gives us the point , which is one of the possible answers.

The easiest way to find the distance between two points whose coordinates are given in the form and is to use the distance formula.

Plugging in the coordinates from our given points, our formula looks as follows

We then simply simplify step by step

Therefore, the distance between the two points is 13.

To find the midpoint, find the midpoint (or just average) for the x and y value separately. For the x-value, this means: . For the y-value, this means: . Thus, the midpoint is (3,4).

Instead of memorizing the distance formula, think of it as a way to use the Pythagorean Theorem. In this case, if you draw both points on a coordinate system, you can draw a right triangle using the two points as corners. The result is a 5-12-13 triangle. Thus, the missing side's length is 13 units. If you don't remember this triplet, then you could use the Pythagorean Theorem to solve.

Whenever you have an angle that is inscribed to the outside edge of a circle and to an angle that passes through the midpoint of the circle, the inscribed angle will always be one half the measurement of the angle that passes through the midpoint of the circle.

Since the angle that passes through the midpoint of the circle is a straight angle (all straight angles measure degrees), the inscribed angle must measure degrees.

Since the sum of the internal angles of all triangles add up to degrees, add up the measurements of the angles that you know and subtract the sum from degrees to find your answer:

If you extend the lines of the parellelogram, you will notice that a parellogram is the same as 2 different sets of parellel lines intersecting one another. When that happens, the following angles are congruent to one another:

Therefore,

In order to solve this problem, you must first solve for what percentage of the entire group comprise of Spanish-speaking students. To do this, divide the total amount of Spanish-speaking students by the total number of students.

Multiply this number by 100 and round up in order to get your percentage.

Then, multiply this number times the total degrees in a circle to find out the measurement of the piece representing Spanish-speaking students on the pie chart.

Round up:

To answer this question, we must put the equation into slope-intercept form, meaning we must solve for . Slope-intercept form follows the format where is the slope and is the intercept.

Therefore, we must solve the equation so that is by itself. First we add to both sides so that we can start to get by itself:

Then, we must subtract from both sides:

We then must divide each side by

Therefore, the slope-intercept form of the original equation is .

The first step is to find the y-coordinates for the two points we are using. To do this we plug our x-values into the equation. Where , we get , giving us the point . Where , we get , giving us the point .

We can now use the distance formula: .

Plugging in our points gives us

To answer this question, we need to find the midpoint of .

To find how far the midpoint of a line is from each end, we use the following equation:

and are taken from the value of the second point and and are taken from the value of the first point. Therefore, for this data:

We can then solve:

Therefore, our midpoint is units between each endpoint's value and unit between each endpoint's value. To find out the location of the midpoint, we subtract the midpoint distance from the point. (In this case it's the point .) Therefore:

So the midpoint is located at

The question asked us what the -coordinate of this point was. Therefore, our answer is .

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

, where is the number of sides.

Therefore, a hexagon like this one has:

.

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure:

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to . Thus, you can write:

Solve for :

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

, where is the number of sides.

Therefore, a hexagon like this one has:

.

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure:

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to . Thus, you can write:

Solve for :

Now, you have to find the largest unknown angle, which is :

The sum of all angles is determined by the following formula for a polygon:

In a pentagon, there are 5 sides, or . Substitute and find the total possible angle in a pentagon.

There are 5 interior angles in a pentagon. Divide the total possible angle by 5 to determine the value of one interior angle.

Each interior angle of a pentagon is 108 degrees.

The sum of three angles in a pentagon is:

Write the formula for the area of a pentagon.

Substitute the side length and simplify the radical.

Write the formula for finding the diagonal of a pentagon.

Plug in the side length and simplify.

Write the diagonal formula for a pentagon.

Substitute the side length and find the diagonal.