### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Do Coordinate Geometry

A deer walks in a straight line for 8 hours. At the end of its journey, the deer is 30 miles north and 40 miles east of where it began. What was the average speed of the deer?

**Possible Answers:**

miles per hour

miles per hour

miles per hour

miles per hour

miles per hour

**Correct answer:**

miles per hour

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:

### Example Question #1 : Coordinate Geometry

Which point satisfies the system and

**Possible Answers:**

None of the other answers

**Correct answer:**

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.

### Example Question #1 : Coordinate Geometry

Find the distance between and

**Possible Answers:**

None of the other answers

**Correct answer:**

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.

### Example Question #1 : Coordinate Geometry

Find the distance between the points and .

**Possible Answers:**

**Correct answer:**

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.

### Example Question #2 : Coordinate Geometry

What is the slope of the line given by the equation ?

**Possible Answers:**

**Correct 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 :

### Example Question #3 : Coordinate Geometry

What is the measurement of ?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #4 : Coordinate Geometry

What is the measurement of ?

**Possible Answers:**

**Correct 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,

### Example Question #5 : Coordinate Geometry

In a poll, Camille learned that of her classmates spoke English at home, spoke Spanish, and spoke other languages. If she were to graph this data on a pie chart, what would be the degree measurement for the part representing students who speak Spanish at home?

**Possible Answers:**

**Correct answer:**

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:

### Example Question #6 : Coordinate Geometry

What is the midpoint between the points and ?

**Possible Answers:**

**Correct answer:**

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).

### Example Question #7 : Coordinate Geometry

Find the distance between the two points and .

**Possible Answers:**

**Correct answer:**

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.

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