SAT Math Exercises – 50 SAT Math Exercises on All Domains
After teaching thousands of SAT exam students, we’ve revealed that a high SAT Math score strongly depends on students’ practice on SAT Math exercises. We’ve seen many students increase their SAT Math scores by around 200 points. This is a huge increase, and it is only possible by doing as many SAT Math exercises and practices as possible.
In this article
šĀ Hint:Ā Do not skip this article, you will findĀ FREE Digital SAT Math Prep resources throughout the article.
Ethan is one of our thousands of successful SAT students.Ā He perfectly scored 1600 on the SAT, the highest score a student can get! We were with thousands of students, like Ethan, in their SAT Math Prep journey. We’ve witnessed how they approach SAT Math Exercises and practices and that is why created this SAT Math Exercices post to help many others!
Before diving into SAT Math Exercises, you must understand the SAT Math structure, question types, and how to approach different types of questions.
The SAT exam consists of two modules, Module I and Module II.Ā Each module consists of 22 questions, and there will be a total of 44 questions. You will have 35 minutes for each module. Going through comprehensive SAT Math Exercises will help you to get a higher score on SAT Math. We prepared the following table to summarize the structure of theĀ SAT Math sections.
Format
Question Types
Stimulus Topics
Test Length
Time Per Module
Number of Questions
Module 1
Two-stage adaptive test design: one Math section administered via two separately timed modules
Discrete; four-option multiple-choice (ā75%) and student-produced response (SPR) (ā25%)
The most critical aspect of the Digital SAT is being adaptive.Ā In SAT Math Module I, you will be asked a broad mix of easy, medium, and hard questions. Then, the difficulty of the SAT Math Module II will depend on your score in the SAT Math Module I. This means that the test āadaptsā to present questions that are more appropriate to a studentās performance level. Going through several SAT Math Exercises and Practices will improve your scores. The following figure depicts the Digital SAT adaptive testing model.
Source: College Board
š”You might be interested in reading theĀ Digital SAT Math Ultimate Guide post. We have provided further details about the SAT Math structure, examples of easy, medium, and hard questions, answers, rationales, and frequently asked questions about the SAT Math.
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Anna B. is one of our thousands of successful SAT students. She scored 800 on SAT Math. You can watch her SAT story.
We’ve created these SAT Math Exercises to provide exercises for the commonly tested concepts on the SAT exam. You can consider enrolling in our Online Digital SAT Math Prep Course for a comprehensive SAT Math Prep.
We’ve listed the SAT Math Exercises for each content domain and skills and knowledge testing points respectively. There are a total of 50 SAT Math Exercises in 10 sets. We do not recommend using a calculator for many of the questions. In case a question needs complex calculations, we mentioned as a note that you can use a calculator for that particular SAT Math Exercise.
Algebra is the first content domain in the SAT Math.Ā The SAT exam will haveĀ around 13 to 15 questions from the Algebra content domain. There are 5 skills and knowledge testing points in the Algebra content domain:
Let’s review each skill and knowledge point and see some SAT Math Exercises for each.
SAT Math Exercise Set 1: Linear Equations in One Variable
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I.Ā
Exercise II.Ā
Exercise III.Ā
Exercise IV.Ā
Exercise V.Ā
Exercise VI.Ā Ethan has $15. He gets dollars from his grandfather. If Ethan has $35 now, what is the value of ?
š”You might be interested in seeing ourĀ Digital SAT Math Practice Test. It covers the hardest questions you may see on the SAT Math exam for each skill and knowledge testing point!
Exercise VI.Ā Ethan has $15. He gets dollars from his grandfather.
Ethan will have dollars.Ā If Ethan has $35 now;
. If we subtract 15 from both sides;
. This yields;
.
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Free Digital SAT Prep Course
SFBS offers a Free Digital SAT Prep Online Course. The course goes through particular skills and knowledge testing points and improves your problem-solving skills and test-taking strategies.
SAT Math Exercise Set 2:Ā Linear Equations in Two Variables
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I.Ā . What is the y-intercept of this graph?
Exercise II.Ā . What is the x-intercept of this graph?
Exercise III.Ā . and are two possible solutions to the equation. What is the value of ?
Exercise IV.Ā . Grapf of line m is given. A line k is parallel to line m. What is the slope of line k?
Exercise V.Ā . If the line h is perpendicular to the given equation’s graph, what is the slope of line h?
Exercise VI.Ā A line passes through (0, 4) and the slope of the line is 2. What is the equation of this line?
Exercise I.Ā The y-intercept of a graph can be found by substituting .
.
. If we divide both sides by 5;
–> (0, 3) is the y-intercept of this graph.
Exercise II.Ā Ā The x-intercept of a graph can be found by substituting .
.
. If we divide both sides by 2;
–> (7, 0) is the x-intercept of this graph.
Exercise III.Ā Ā We can find the k and l values by substituting the corresponding x and y values in the equation.
When x = 7, y = k;
. This yields;
. If we subtract 35 from both sides;
. This yields;
If we divide both sides by 7;
.
When x = l, y = 5;
. This yields;
. If we subtract 35 from both sides;
. This yields;
If we divide both sides by 5;
.
We found k = 5, and l = 7;
Exercise IV. We should write the equation in slope-intercept form. So, we should isolate the y;
. Subtract from both sides. This yields;
. If we divide both sides by 3;
. This equation yields;
. This equation is in the slope-intercept form y = mx + b.
. Since the line k is parallel to this line, it will have the same slope as well.
Exercise V.Ā Ā We should write the equation in slope-intercept form. So, we should isolate the y;
. Subtract from both sides. This yields;
. If we divide both sides by 4;
. This equation yields;
. This equation is in the slope-intercept form y = mx + b.
. If line h is perpendicular to this line, the products of the slopes of these two lines will be -1. If we say for the slope of line h;
. If we divide both sides by ;
. This yields;
Exercise VI.Ā In the slope-intercept form, y = mx + b, m is the slope, and (0, b) is the y-intercept. It’s given that the line passes through (0, 4) and the slope of the line is 2. This means b = 4, and m = 2. If we substitute these in the equation;
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SAT Math Exercise Set 3: Linear Functions
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I.Ā Ā The graph of y = f(x) passes through the points (1, 4) and (3, 10). What is the function f ?
Exercise II.Ā Ā The function f is defined by the equation . What is the value of f(x) when ?
Exercise III. An electrician charges a $40 fixed fee plus $25 per hour. If the function h models the total fee for the electrician for t hours of work, write the function h.
Exercise IV. The function f is defined by . What is the y-intercept of the graph of in the xy-plane?
Exercise V. The graph of y =Ā f(x) and y=h(x)Ā are perpendicular to each other in the xy-plane. Function f is defined by . If , write the y=h(x) function.
Exercise VI. The function models the altitude of an airplane m minutes after takeoff. According to the model, what is the altitude of the airport where the airplane took off?
Exercise I. We can find the line equation if we know two points where the line passes through. We can find the slope with the following formula:
. If we substitute the x and y coordinates of the two points (1, 4) and (3, 10);
. This yields;
.
In the slope-intercept form, y = mx + b. If we substitute m = 3 and one of the points coordinates in this equation we can find the value of b as well.
. We substituted (1, 4). x = 1, y = 4. Solving this equation yields;
. Now, we can write the equation;
Exercise II. We should find the value of x when .
. If we divide both sides by 3;
. If we substitute x = 2 in the function equation;
. This yields;
Exercise III.Ā It’s given that $40 is a fixed fee. So, when t=0, the function h(0)=40. In other words, the y-intercept is (0, 40).
For every t hours of work, the electrician will charge 25t dollars. In other words, the slope is 25. Therefore;
Exercise IV.Ā The given function is already in the slope-intercept form: y = mx + b. Therefore;
(0, -7) is the y-intercept of the .
Exercise V.Ā If the y =Ā f(x) and function y=h(x) are perpendicular to each other in the xy-plane, the product of the slopes of these two lines is (-1).
. It is in slope-intercept form of y = mx + b. Therefore;
. If we say the slope of y=h(x) is ;
. If we divide both sides by 2;
. It’s given that , if we substitute and (6, 2), x = 6 and y=2 in the slope-intercept form of function h;
. This yields;
. If we add 3 on both sides;
. We found and b = 5. Therefore;
Exercise VI.Ā It’s given that gives the altitude of the airplane, in feet, m minutes after take-off. Therefore, when m=0, the airline is still on the runway and a(0) equals the airport’s altitude.
. This yields;
feet.
SAT Math Exercise Set 4:Ā Systems of Two Linear Equations in Two Variables
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I.Ā
The solution to the given system of equations is (x, y). What is the value of y)
Exercise II.Ā
For the given system of equations, what is the value of y ?
Exercise III.Ā
The solution to the given system of equations is (x, y). What is the value of x + y?
Exercise IV.Ā Ā
In the given system of equations, k is a constant. If the system has no solution, what is the value of k?
Exercise I.Ā Ā If we multiply both sides of the first equation by 3;
. This yields;
. In the second equation, we can substitute 6x for -9y
. This yields;
. Dividing both sides by 2;
and this yields;
.
Exercise II. If we subtract 7 from both sides of the equation;
. This yields;
. Now, we can substitute x = 4 in the second equation;
. This yields;
Exercise III.Ā We can add the left side of the 1st equation to the left side of the second equation, and the right side of the 1st equation to the right side of the second equation.
. If we open the parentheses; and combine the like terms;
. This yields;
. If we use the common factor of 3 on the left side of the equation;
. If we divide both sides by 3; we will find the value of x + y.
. This yields;
Exercise IV. For two systems of equations in the form of:
, and
x and y have no solution if the lines are parallel and distinct. For these two lines to be parallel;
and for these lines to be distinct;
ā and ā
Ā –> A=2, B=5 and C=12. –> D=6, E=k and F=33.
For these two lines to be parallel;
. If we substitute the values;
. If we do cross-multiplication;
. Dividing both sides by 2 yields;
. Note that, when k=15, these two lines are parallel. For them to be distinct;
ā –> This is true. And
ā –> this is true as well.
So, when k = 15, the given system of equations will have no solutions.
SAT Math Exercise Set 5:Ā Linear Inequalities in One or Two Variables
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I. Ryan has $100 and wants to purchase balls. A store sells basketballs and baseballs. Basketballs are $15 each and baseballs are $20 each.Ā If x represents the number of basketballs and y represents the number of baseballs Ryan can purchase, what is the inequality representing this situation?
Exercise II.Ā The maximum value of is 13 greater than another number k. What is the inequality for x in terms of k?
Exercise III. A High School has students whose height is between 160 cm to 186cm. If h represents the height of a student in this High School, what is the inequality representing the height of a student?
Exercise IV.
(2, p) is a solution to the given system of inequalities. What is the maximum integer value for p?
Exercise I.Ā It’s given that x represents the number of basketballs and y represents the number of baseballs Ryan can purchase. Basketballs are $15 each and baseballs are $20 each. Therefore;
= Total cost of basketballs
= Total cost of baseballs.
If we sum up the cost of basketballs and baseballs Ryan can purchase;
= Total cost of all balls Ryan can purchase.
Ryan has $100, so, he can spend a maximum of $100 for all balls. Therefore; the total cost of all balls Ryan can purchase must be equal to or less than 100.
Exercise II.Ā 13 greater than a number k is;
.
If the maximum value of is 13 greater than another number k, is less than or equal to .
. If we subtract 3 from both sides of the inequality;
. This yields;
Exercise III.Ā If the student’s height is between 160 and 186 cm, a student’s height must be equal to or greater than 160 and less than or equal to 186cm.
Exercise IV.Ā If we multiply both sides of the first equation by (-1) inequality sign will reverse.
. This yields;
.
If we divide both sides of this inequality by 5;
. This yields;
.
If we divide both sides of the second inequality by 2;
. This yields;
.
As we have both inequalities in terms of y; we can combine them as follows;
.
It’s given that (2, p) is a solution to the system of inequalities. If we substitute x = 2, and y = p in the inequality;
. This yields;
, or
The maximum integer value for p is .
SAT Math Exercises for Advanced Math
Advanced Math is the second content domain in SAT Math. In the SAT exam, there will be around 13 to 15 questions from the Advanced Math content domain. There are 3 skills and knowledge testing points in the Advanced Math content domain:
Equivalent expressions
Nonlinear equations in one variable and systems of equations in two variables
. We need to isolate p to write the p-value in terms of x and y. Divide both sides of the equation by 221x;
. This yields;
.
Exercise II.Ā If we multiply both sides of the given equation by ;
. This yields;
. If we substitute the x value;
.
Exercise III. You should know that , and;
. With the help of these two, we can rewrite the left side of the equation as follows;
. This is the left side of the equation.
We can rewrite the right side of the equation as follows;
. This yields;
. This yields;
. This is the right side of the equation.
Now, combine together the left and right sides of the equation;
. The bases of the right and left sides of the equation are the same now. Therefore, powers must be the same as well.
. If multiply both sides by 3;
. This yields;
. If we subtract 5x+3 from both sides;
. This yields;
.
Exercise IV.Ā We can rewrite the first fraction’s numerator as follows;
. Therefore, the first fraction can be rewritten as;
. There are (x-2) both in the numerator and denominator, so we can eliminate them, and the first fraction yields;
. This is the simplified expression for 1st fraction.
There are two expressions in the numerator of the 2nd fraction. We can rewrite the first expression as follows;
. Therefore the numerator can be rewritten as follows;
.
We can rewrite an equation in the form of as . Therefore, we can rewrite the denominator of the 2nd fraction as follows;
. If we rewrite the 2nd fraction with the expressions we found;
. We see that (x-2)(x+2) are present both in the numerator and denominator of the 2nd fraction. Therefore, we can eliminate them.
. This is the simplified expression for 2nd fraction.
If we sum up the simplified expressions for the 1st and 2nd fractions;
. The result yields;
.
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SAT Math Exercise Set 7:Ā Nonlinear Equations in One Variable and Systems of Equations in Two Variables
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I. Two variables, x, and y are related such that for each increase of 1 in the value of x, the value of y increases by a factor of 5. When x=0, y=10. Write the y in terms of x.
Exercise II. .
One solution to the given equation can be written as , where k is a constant. What is the value of k?
Exercise III..
In the given equation, p is a constant. The equation has exactly one solution. What is the value of p?
Exercise IV. .
A solution to the given system of equations is (x, y), where x>0. What is the value of x?
Exercise I.Ā If for each increase of 1 in the value of x, the value of y increases by a factor of 5 there should be an exponential relationship between x and y. We can write the relationship as follows;
. a is a constant. Let’s test this.
When x = 1;
When x = 2; 5 times 5a equals to 25a. Therefore, the equation is correct.
It’s given that when x=0, y=10. If we substitute x and y values in our equation, we can find the a value.;
. The zero power of a number is equal to 1. Therefore;
. This yields;
Exercise II. In the form of an equation , the solution to the equation is as follows;
.
If we write the given equation in the form of , we can find the and values.
If we subtract 11 from both sides of the given equation;
. This yields;
. This is in the form of where a = 2, b = -8 and c = -11. If we substitute the values in the solution equation;
. This yields;
. This yields;
. This yields;
± . Therefore;
and .
It’s given that one of the solutions is . Therefore;
Exercise III.Ā In the form of an equation ;
is called discriminant. If the discriminant of an equation;
. There are two and solutions.
. There is exactly one solution x.
. There are no real solutions.
is in the form of , where a = -3, b = p and c = -12. For this equation to have exactly one solution, the discriminant should be zero. (). If we substitute the a, b, and c values in the discriminant;
. This yields;
. If we add 144 on both sides;
. If we apply the square root on both sides;
. This yields;
Exercise IV.Ā If we divide both sides of the first equation by 2;
. This yields;
. If we substitute this in the second equation for y;
. If we subtract from both sides;
. This yields;
.
We can rewrite an equation as . Therefore, we can rewrite the equation as;
.
If , then . This yields;
SAT Math Exercise Set 8:Ā Nonlinear Functions
To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
Exercise I.
The given equation defines the function f. What is the minimum value of f(x)?
Exercise II.Ā The function f is defined by . What is the value of f(6)?
Exercise III.Ā The function gives a spring’s length, in feet, when an object of w kilograms is hung, where . What is the best interpretation of 15 in this context?
Exercise I. For a parabola written in the form of , if a > 0, then the graph is upward. In the given equation, a = 3. So, the minimum value of the f(x) will be its vertex.
If we can rewrite the given equation in the form of where a, h, and k are constants, (h,k) is the vertex point.
If we use the common factor of 3, we can rewrite the given equation as follows;
We can rewrite an equation as . Therefore, we can rewrite the equation as;
. We can rewrite as . Therefore;
. If we expand the parenthesis;
. This is in the form of where a = 3, h = 1 and k = 12. The vertex point is (h, k) = (1, 12).
Exercise II. It’s given that .
. This yields x = 2. If we substitute x = 2 in the function equation;
. This yields;
. This yields;
.
Exercise III.Ā When w = 0;
. So, when there are no objects, the length of the spring is 15 feet. We can conclude that the initial length of the spring when no object is hung is 15 feet.
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SAT Math Exercises for Problem-Solving and Data Analysis
Problem-solving and Data Analysis is the third content domain in SAT Math. In the SAT exam, there will be around 5 to 7 questions from the Problem-solving and Data Analysis content domain. There are 7 skills and knowledge testing points in the Problem-solving and Data Analysis content domain:
Ratios, rates, proportional relationships, and units
Percentages
One-variable data: distributions and measures of center and spread
Two-variable data: models and scatterplots
Probability and conditional probability
Inference from sample statistics and margin of error
Evaluating statistical claims: observational studies and experiments
We’ve listed one SAT Math Exercise for each skill and knowledge testing below. To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
SAT Math Exercise Set 9: Problem-Solving and Data Analysis
Exercise I. Alisa purchased a box of 100 tea bags. She uses one tea bag for each cup of tea. If Alisa drinks 3 cups of tea every day, in how many days will the number of tea bags in the box drop below 20?
Exercise II.Ā A store offers a 20% discount on a certain bag. During the Black Friday promotion, an additional 10% discount is applied on all products in the store. If the final price of the bag is x % of the initial price, what is the value of x?
Exercise III.Ā 1, 3, 7, 7, 8, 5, 2, 11
What is the sum of the median and mean of the data set shown?
Exercise IV.Ā
The scatterplot shows the relationship between two variables, x and y. A line of best fit for the data is also shown. What is the difference between the y-coordinate of the data point with x = 4 and the y-value predicted by the line of best fit at x = 4?
Exercise V.Ā The following table shows the number of students in each grade in a High School.
Grades
Number of Students
9
105
10
101
11
100
12
72
25% of the Grade 12 students attend French club. If a student is picked randomly, what is the probability of selecting a Grade 12 student not attending the French club?
Exercise VI.Ā A random sample of 60 people from a town with a population of 18,756 were asked for their opinion on a recent government policy. If 34 people in the sample support the government policy, what is the expected number difference between the supporters and non-supporters in the town?
Note: You can use a calculator in this SAT Math Exercise
Exercise VII.Ā A study is conducted in the state of Utah. A sample of people over 50 years old are asked how many times they visit a doctor each year. What is the largest population to which the result of the survey can be generalized?
Exercise I.Ā It’s given that Alisa uses one tea bag per cup and she drinks 3 cups of tea every day. This means she uses 3 tea bags every day. If we say that in x days, the number of tea bags in the box will drop below 20, she will use 3x the number of tea bags until that day. So, the inequality will be as follows;
. If we add 3x to both sides and subtract 20 from both sides of the equation;
. This yields;
. If we divide both sides by 3;
. This yields;
. The smallest integer greater than 26.66 is 27. Therefore, in 27 days, the number of tea bags in Alisa’s tea box will drop below 20.
Exercise II.Ā Let’s say the initial price of the bag is 100p. If a 20% discount is applied to this bag;
On Black Friday, an additional 10% discount is applied.
Discount on Black Friday =
Final Price = Discounted Price – Discount on Black Friday
If the final price of the bag is x % of the initial price;
. If we substitute the values;
If we divide both sides by p, this yields;
Exercise III.Ā First, we should arrange the data set in increasing order as follows;
1, 2, 3, 5, 7, 7, 8, 11
The median of a data set is the middle value when the data points are arranged in order. If there is an even number of data points, the median is the average of the two middle numbers. There are 8 number of data points in our data set. Therefore, the average of the 4th and 5th data points will be the median.
. If we substitute the values;
. This yields;
. We found the median.
The mean of a data set is the average of all data points.
. If we substitute the values;
. This yields;
. We found the mean.
The sum of the median and mean is;
Exercise IV.Ā We can draw the y-value of the data point when x = 4 and the y-value predicted by the best line of fit as follows;
The red lines show the y-value of that data point when x = 4, y = 7
The blue lines show the y-value predicted by the best line of fit when x = 4, y = 7.5 (in the middle of 7 and 8)
, or
Exercise V.Ā First, we should find the number of Grade 12 students who are not attending the French club. It’s given that 25% of the Grade students attend French club. This means;
of Grade 12 students do not attend French club. There are 72 Grade 12 students. Therefore;
54Grade 12 students do not attend the French club.
Now, we should find the total number of students in the High School. It will be the sum of all the number of students in Grades 9, 10, 11, and 12.
Total Number of Students in High School =
The probability of picking a Grade 12 student who is not attending the French club is;
Exercise VI.Ā It’s given that the sample size is 60 people. 34 out of 60 people support the government policy. Therefore;
Non-supporters in the sample = . The difference between the supporters and non-supporters in the sample group is;
Difference Between Supporters and Non-Supporters in the Sample =
We expect that the ratio of the difference between supporters and non-supporters in the sample will be the same in the town. If we say the difference between supporters and non-supporters in the town is x;
. If we do cross-multiplication;
. If we divide both sides by 60;
. We can conclude that the number of supporters will be around 2,501 greater than the number of non-supporters.
Exercise VII.Ā It’s given that the study is conducted in Utah and over 50 years old people are asked. Therefore, the largest population that this survey can be generalized is “People older than 50 years old in the State of Utah”.
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SAT Math Exercises for Geometry and Trigonometry
Geometry and Trigonometry is the fourth and last content domain in SAT Math.Ā The SAT exam will haveĀ around 5 to 7 questions from the Geometry and Trigonometry content domain. There are 4 skills and knowledge testing points in the Geometry and Trigonometry content domain:
Area and Volume
Lines, angles, and triangles
Right triangles and trigonometry
Circles
š”You might be interested in seeing our SAT Geometry and Trigonometry post. It covers the hack points you should know and SAT Geometry Practice Questions!
We’ve listed one SAT Math Exercise for each skill and knowledge testing below. To improve your math skills, we do not recommend using a calculator when solving these SAT Math Exercises.
SAT Math Exercise Set 10: Geometry and Trigonometry
Exercise I.Ā One side of a rectangle and square are common. The area of the rectangle is two times the area of the square. If the perimeter of the rectangle is 10 units greater than the perimeter of the square, what is the length of the rectangle in units?
Exercise II.
Note: Figure not drawn to scale.
In the figure, line m is parallel to line n, and line k intersects both lines. What is the value of x + y ?
Exercise III.Ā One leg of an isosceles right triangle A is common with the shortest leg of another right triangle B. The length of the longest side of the triangle B is 17, and longer leg length is 15. What is the length of the longest side of triangle A?
Exercise IV. The graph of in the xy-plane is a circle. What is the area of the circle?
Note: You can use a calculator in this SAT Math Exercise
Exercise I.Ā Let’s draw the expressed shapes. It’s given that one side of a rectangle and a square are common. If we say “w” to one side of the square, and “l” to the other side of the rectangle, our figure will be as follows.
The area of the square =
The area of the rectangle =
It’s given that the area of the rectangle is two times the area of the square;
. If we divide both sides by w;
. This yields;
. We found the relationship between the length and width of the rectangle.
The perimeter of the square is =
The perimeter of the rectangle is =
It’s given that the perimeter of the rectangle is 10 units greater than the perimeter of the square. This yields;
. We can substitute and this yields;
. This yields;
. If we subtract 2l from both sides;
. This yields;
Exercise II.Ā Since the lines m and n are parallel;
angle next to y° will be x° as well. The sum of angles on a line is 180°. Therefore, x+y = 180°.
Exercise III. Triangle A is an isosceles right triangle. This means the lengths of its legs are the same. It is a special right triangle with the angles 45°-45°-90°. It’s given that one of the legs of Triangle A is common with the shortest leg of Triangle B, the length of the longest side of Triangle B is 17, this means it is the hypotenuse, and the longer leg length is 15, which is next leg to the common leg with Triangle A.
Let’s say the length of one leg of the isosceles right triangle is a. Sine of an angle is;
. For an isosceles right triangle, the angles will be 45°-45°-90°.
. Therefore;
. If we do cross-multiplication;
. If we divide both sides by ;
. This yields;
. This yields;
If we draw the described figure, it will be as follows;
In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. Therefore;
. If we isolate the ;
. This yields;
. If we apply square root on both sides;
. This yields;
.
The length of the longest side of Triangle A is its hypotenuse and it is . If we substitute the value of a;
Exercise IV. The standard form of a circle is where (h,k) is the center of the circle and r is the radius. Therefore, we should rewrite the given equation in the standard circle form to find its radius, r.
We can rewrite an equation as . Therefore, we can rewrite;
. If we subtract from both sides;
.
Similarly, we can rewrite the part as follows;
.
If we substitute these in the given equation;
. This yields;
. This yields;
. This is now in the standard form of a circle , where , and .
If we apply square root on both sides of the r equation;
. This yields;
.
Area of a Circle = . If we substitute the values in this formula;
Area of the Circle = . This yields;
Area of the Circle = . (You can round this to 254.47.)
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