Geometry - ACT Math
Card 0 of 7119
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?
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?
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 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:
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What is the slope of the line given by the equation 
?
What is the slope of the line given by the equation
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 
:
To find the slope, put the line in slope intercept form. In other words put the equation in
From here we can see our slope equals
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Find the distance between 
and 
Find the distance between
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. 
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 =
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.
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Which point satisfies the system 
and 
Which point satisfies the system
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.
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
Now that we have two possible values for x, we can plug each value into either equation to obtain two values for y. For
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Find the distance between the points 
and 
.
Find the distance between the points
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.
The easiest way to find the distance between two points whose coordinates are given in the form
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.
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What is the midpoint between the points 
and 
?
What is the midpoint between the points
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).
To find the midpoint, find the midpoint (or just average) for the x and y value separately. For the x-value, this means:
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Find the distance between the two points 
and 
.
Find the distance between the two points
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.
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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What is the measurement of 
?
What is the measurement of
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:
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
Since the sum of the internal angles of all triangles add up to
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What is the measurement of 
?
What is the measurement of
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, 
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,
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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?
In a poll, Camille learned that
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:
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:
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Which of the following is the slope-intercept form of 
?
Which of the following is the slope-intercept form of
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 
.
To answer this question, we must put the equation into slope-intercept form, meaning we must solve for
Therefore, we must solve the equation so that
Then, we must subtract
We then must divide each side by
Therefore, the slope-intercept form of the original equation is
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Following the line 
, what is the distance from the the point where 
to the point where 
?
Following the line
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 
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 can now use the distance formula:
Plugging in our points gives us
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The coordinates of the endpoints of 
, in the standard 
coordinate plane, are 
and 
. What is the 
-coordinate of the midpoint of ?
The coordinates of the endpoints of
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 
.
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:
We can then solve:
Therefore, our midpoint is
So the midpoint is located at
The question asked us what the
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If 
, 
, and 
, what is the measure, in degrees, of 
?
If 
The question states that 
. The alternate interior angle theorem states that if two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent; therefore, we know the following measure:
The sum of angles of a triangle is equal to 180 degrees. The question states that 
; therefore we know the following measure:
Use this information to solve for the missing angle: 
The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:
The measure of 
is 148 degrees.
The question states that 
The sum of angles of a triangle is equal to 180 degrees. The question states that 
Use this information to solve for the missing angle:
The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:
The measure of 
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In the following diagram, lines 
and 
are parallel to each other. What is the value for 
?
In the following diagram, lines
When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.
Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.
When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.
Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.
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Two pairs of parallel lines intersect:
If A = 135o, what is 2*|B-C| = ?
Two pairs of parallel lines intersect:
If A = 135o, what is 2*|B-C| = ?
By properties of parallel lines A+B = 180o, B = 45o, C = A = 135o, so 2*|B-C| = 2* |45-135| = 180o
By properties of parallel lines A+B = 180o, B = 45o, C = A = 135o, so 2*|B-C| = 2* |45-135| = 180o
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Lines A and B in the diagram below are parallel. The triangle at the bottom of the figure is an isosceles triangle.
What is the degree measure of angle 
?
Lines A and B in the diagram below are parallel. The triangle at the bottom of the figure is an isosceles triangle.
What is the degree measure of angle
Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles, 
and 
which will sum up to 
. Setting up an algebraic equation for this, we get 
. Solving for 
, we get 
. With this, we can get either 
(for the smaller angle) or 
(for the larger angle - must then use supplementary rule again for inner smaller angle). Either way, we find that the inner angles at the top are 80 degrees each. Since the sum of the angles within a triangle must equal 180, we can set up the equation as 
degrees.
Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles,
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Figure not drawn to scale.
In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?
Figure not drawn to scale.
In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?
Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.
Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:
x + (x + 81) + x = 180
Simplify by collecting the x terms.
3x + 81 = 180
Subtract 81 from both sides.
3x = 99
Divide by 3.
x = 33.
This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.
measure of CPB = 33 + 33 = 66.
The answer is 66.
Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.
Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:
x + (x + 81) + x = 180
Simplify by collecting the x terms.
3x + 81 = 180
Subtract 81 from both sides.
3x = 99
Divide by 3.
x = 33.
This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.
measure of CPB = 33 + 33 = 66.
The answer is 66.
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One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?
One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?
Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.
Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180.
We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:
(1/2)y = 2x.
Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.
(1/2)(180-x) = 2x.
Multiply both sides by 2 to get rid of the fraction.
(180 – x) = 4x.
Add x to both sides.
180 = 5x.
Divide both sides by 5.
x = 36.
The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:
x + z = 90.
Now, we can substitute 36 as the value of x and then solve for z.
36 + z = 90.
Subtract 36 from both sides.
z = 54.
The answer is 54.
Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.
Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180.
We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:
(1/2)y = 2x.
Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.
(1/2)(180-x) = 2x.
Multiply both sides by 2 to get rid of the fraction.
(180 – x) = 4x.
Add x to both sides.
180 = 5x.
Divide both sides by 5.
x = 36.
The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:
x + z = 90.
Now, we can substitute 36 as the value of x and then solve for z.
36 + z = 90.
Subtract 36 from both sides.
z = 54.
The answer is 54.
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In the diagram, AB || CD. What is the value of a+b?
In the diagram, AB || CD. What is the value of a+b?
Refer to the following diagram while reading the explanation:
We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection). Therefore, it is 20°.
Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°. Based on the rule for supplements, we know that a + 40° = 180°. Solving for a, we get a = 140°.
Therefore, a + b = 140° + 20° = 160°
Refer to the following diagram while reading the explanation:
We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection). Therefore, it is 20°.
Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°. Based on the rule for supplements, we know that a + 40° = 180°. Solving for a, we get a = 140°.
Therefore, a + b = 140° + 20° = 160°
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