pythagorean theorem converse and inequalities assignment

Become a math whiz with AI Tutoring, Practice Questions & more.

The Converse of Pythagorean Theorem

We assume you're familiar with the Pythagorean Theorem .

The converse of the Pythagorean Theorem is:

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

That is, in Δ A B C , if c 2 = a 2 + b 2 then ∠ C is a right triangle, Δ P Q R being the right angle.

We can prove this by contradiction.

Let us assume that c 2 = a 2 + b 2 in Δ A B C and the triangle is not a right triangle.

Now consider another triangle Δ P Q R . We construct Δ P Q R so that P R = a , Q R = b and ∠ R is a right angle.

By the Pythagorean Theorem, ( P Q ) 2 = a 2 + b 2 .

But we know that a 2 + b 2 = c 2 and a 2 + b 2 = c 2 and c = A B .

So, ( P Q ) 2 = a 2 + b 2 = ( A B ) 2 .

That is, ( P Q ) 2 = ( A B ) 2 .

Since P Q and A B are lengths of sides, we can take positive square roots.

That is, all the three sides of Δ P Q R are congruent to the three sides of Δ A B C . So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since Δ A B C is congruent to Δ P Q R and Δ P Q R is a right triangle, Δ A B C must also be a right triangle.

Check whether a triangle with side lengths 6 cm, 10 cm, and 8 cm is a right triangle.

Check whether the square of the length of the longest side is the sum of the squares of the other two sides.

( 10 ) 2 = ? ( 8 ) 2 + ( 6 ) 2     100 = ? 64 + 36   100 = 100

Apply the converse of Pythagorean Theorem.

Since the square of the length of the longest side is the sum of the squares of the other two sides, by the converse of the Pythagorean Theorem, the triangle is a right triangle.

A corollary to the theorem categorizes triangles in to acute, right, or obtuse.

In a triangle with side lengths a , b , and c where c is the length of the longest side,

if c 2 < a 2 + b 2 then the triangle is acute, and

if c 2 > a 2 + b 2 then the triangle is obtuse.

Check whether the triangle with the side lengths 5 , 7 , and 9 units is an acute, right, or obtuse triangle.

The longest side of the triangle has a length of 9 units.

Compare the square of the length of the longest side and the sum of squares of the other two sides.

Square of the length of the longest side is 9 2 = 81 sq. units.

Sum of the squares of the other two sides is

5 2 + 7 2 = 25 + 49                               = 74  sq . units

That is, 9 2 > 5 2 + 7 2 .

Therefore, by the corollary to the converse of Pythagorean Theorem, the triangle is an obtuse triangle.

• Canadian Politics Tutors
• Horizon Zero Dawn Tutors
• Certified ScrumMaster Courses & Classes
• Computer Science Tutors
• Iowa Bar Exam Courses & Classes
• CDR - Commission on Dietetic Registration Test Prep
• College World History Tutors
• IB Sports, Exercise and Health Science SL Tutors
• ACT Science Test Prep
• ISEE Tutors
• French 4 Tutors
• AP Geography Tutors
• Analysis Tutors
• Series 31 Courses & Classes
• WEST-E Test Prep
• ACCUPLACER Arithmetic Tutors
• Parapsychology Tutors
• GRE Courses & Classes
• Geometry Tutors
• SAT Subject Test in World History Courses & Classes
• Tulsa Tutoring
• Sacramento Tutoring
• Dallas Fort Worth Tutoring
• St. Louis Tutoring
• Pittsburgh Tutoring
• Washington DC Tutoring
• Boston Tutoring
• Orlando Tutoring
• Raleigh-Durham Tutoring
• Austin Tutoring
• ACT Tutors in Phoenix
• LSAT Tutors in Miami
• GRE Tutors in Boston
• Math Tutors in Chicago
• Reading Tutors in New York City
• SAT Tutors in Boston
• English Tutors in Boston
• ACT Tutors in Miami
• Spanish Tutors in Phoenix
• Reading Tutors in Chicago

The Converse of the Pythagorean Theorem

In these lessons, we will learn

• the converse of the Pythagorean Theorem
• how to use the converse to determine whether a triangle is acute, right or obtuse
• how to prove the converse of the Pythagorean Theorem

Related Pages Pythagorean Theorem Pythagorean Theorem Word Problems Applications Of Pythagorean Theorem More Geometry Lessons

The following figures show the Converse of the Pythagorean Theorem. Scroll down the page for more examples, solutions, and proofs of the Converse of the Pythagorean Theorem.

The Pythagorean Theorem states that

In any right triangle, the sum of the squared lengths of the two legs is equal to the squared length of the hypotenuse.

The converse of the Pythagorean Theorem states that

For any triangle with sides a, b, c, if a 2 + b 2 = c 2 , then the angle between a and b measures 90° and the triangle is a right triangle.

How to use the converse to determine the type of triangle

We can also use the converse of the Pythagorean theorem to check whether a given triangle is an acute triangle, a right triangle or an obtuse triangle.

For a triangle with sides a, b and c and c is the longest side then:

If c 2 < a 2 + b 2 then it is an acute triangle, i.e. the angle facing side c is an acute angle.

If c 2 = a 2 + b 2 then it is a right triangle, i.e. the angle facing side c is a right angle.

If c 2 > a 2 + b 2 then it is an obtuse triangle, i.e. the angle facing side c is an obtuse angle.

Example: Determine whether a triangle with sides 3 cm, 5 cm and 7 cm is an acute, right or obtuse triangle.

Solution: We choose the two shorter sides to be a and b and the longest side to be c. So a = 3, b = 5 and c = 7. a 2 + b 2 = 3 2 + 5 2 = 9 + 25 = 34 c 2 = 7 2 = 49 49 > 34 → c 2 > a 2 + b 2 , and so the triangle is an obtuse triangle.

Example: Determine whether a triangle with sides 12 cm, 14 cm and 18 cm is an acute, right or obtuse triangle.

Solution: We choose the two shorter sides to be a and b and the longest side to be c. So a = 12, b = 14 and c = 18. a 2 + b 2 = 12 2 + 14 2 = 144 + 196 = 340 c 2 = 18 2 = 324 340 < 34 → c 2 < a 2 + b 2 , and so the triangle is an acute triangle.

Example: Determine whether a triangle with sides 8 cm, 15 cm and 17 cm is an acute, right or obtuse triangle.

Solution: We choose the two shorter sides to be a and b and the longest side to be c. So a = 8, b = 15 and c = 17. a 2 + b 2 = 8 2 + 15 2 = 64 + 225 = 289 c 2 = 17 2 = 289 289 = 289 → c 2 = a 2 + b 2 , and so the triangle is an right triangle.

How to use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle? Triples to memorize 3-4-5, 5-12-13, 8-15-17, 7-24-25

How to use the Pythagorean Theorem and its Converse to determine if a triangle is acute, right, or obtuse? • Use the Pythagorean Theorem to determine if a triangle is acute, right, or obtuse. • Use the triangle inequality to determine if a triangle can be formed. The triangle inequality states that in order to construct a triangle, the sum of the shorter sides must be greater than the longest side.

Examples: Determine if the lengths represent the sides of an acute, right, or obtuse triangle, if a triangle is possible.

The Converse of the Pythagorean Theorem This video discusses the converse of the Pythagorean Theorem and how to use it verify if a triangle is a right triangle. Also, two triangle inequalities used to classify a triangle by the lengths of its sides. Both are related to the Pythagorean Theorem.

Example: Determine if √13, 10 and 12 make a right triangle.

How to Use the Converse of the Pythagorean Theorem? How to determine if three given lengths of the sides of a triangle make a right triangle? This is an application of the Converse of the Pythagorean Theorem.

Example: The numbers represent the lengths of the sides of a triangle. Classify each triangle as acute, obtuse, or right. a) 15,20,25 b) 10,15,20

Converse of Pythagorean Theorem

Specify all values of x that make the statement true. a) ∠1 is an obtuse angel. b) The triangle is isosceles. c) No triangle is possible.

Explain why x must equal 5.

Explain why ∠D must be a right angle.

Explain why ∠P must be a right angle.

Proof of the Converse of the Pythagorean Theorem The following video will show a proof of the Converse of the Pythagorean Theorem.

Proof of the Converse of Pythagoras' Theorem

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Converse of Pythagoras Theorem

The converse of Pythagoras theorem is the reverse of the Pythagoras theorem and it helps in determining if a triangle is acute, right, or obtuse if the sum of the squares of two sides of a triangle is compared to the square of its third side. The Pythagorean theorem is the most used in trigonometry. Let us learn more about the converse of the Pythagoras theorem, the proof, and solve a few examples.

What is the Converse of Pythagoras Theorem?

The converse of Pythagoras theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. The converse is the complete reverse of the Pythagoras theorem. The main application of the converse of the Pythagorean theorem is that the measurements help in determining the type of triangle - right, acute, or obtuse. Once the triangle is identified, constructing that triangle becomes simple. There are three cases that occur:

1. If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle .

2. If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle .

3. If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle .

Pythagoras Theorem

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the given triangle ABC, we have BC 2 = AB 2 + AC 2 ​​. Here, ​​​​AB is the base, AC is the altitude or the height, and BC is the hypotenuse. In other words, we can say, in a right triangle, (Opposite) 2 + (Adjacent) 2 = (Hypotenuse) 2 ​​​​​​.

Proof of Converse of Pythagoras Theorem

Statement: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Proof: Here, we are given a triangle ABC in which AC 2 = AB 2 + BC 2 . We need to prove that ∠B = 90°.

To start with, we construct a ΔPQR right-angled at Q such that PQ = AB and QR = BC.

Now, from Δ PQR, we have:

PR 2 = PQ 2 + QR 2 (Pythagoras Theorem, as ∠Q=90°)

or PR 2 = AB 2 + BC 2 (By construction)........ (1)

But AC 2 = AB 2 + BC 2 (Given).......... (2)

So, AC = PR (From (1) and (2)).............. (3)

Now, in ΔABC and ΔPQR,

AB = PQ (By construction)

BC = QR (By construction)

AC = PR (Proved in (3))

So, ΔABC ≃ ΔPQR (According to the SSS congruence)

∠B = ∠Q (Corresponding angles of congruent triangles)

∠Q = 90° (By construction)

So ∠B = 90°.

Hence, the converse of the Pythagoras theorem is proved.

Converse of Pythagoras Theorem Formula

The converse of Pythagoras theorem formula is c 2 = a 2 + b 2 , where a, b, and c are the sides of the triangle.

Related Topics

Listed below are a few topics related to the converse of the Pythagoras theorem, take a look.

• Right Triangle Formulas
• Hypotenuse Leg Theorem
• What is Similarity
• Similarity in Triangles

Examples on Converse of Pythagoras Theorem

Example 1: The side of the triangle is of length 8 units, 10 units, and 6 units. Is this triangle a right triangle? If so, which side is the hypotenuse?

We know that the hypotenuse is the longest side in a triangle. The side or lengths is given as 8 units, 10 units, and 6 units. Therefore, 10 units is the hypotenuse.

Using the converse of Pythagoras theorem, we get,

(10) 2 = (8) 2 + (6) 2

100 = 64 + 36

Since both sides are equal, the triangle is a right triangle.

Example 2: Check if the triangle is acute, right, or an obtuse triangle with side lengths as 6, 8, and 11 units.

Solution: According to the length, we know that 11 units are the longest side.

Compare the square lengths of both the sides in the equation c 2 = a 2 + b 2 .

(11) 2 = (6) 2 + (8) 2

121 = 36 + 64

Hence, (11) 2 > (6) 2 + (8) 2

Therefore, according to the application of converse of Pythagoras theorem (If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle), the triangle is an obtuse triangle.

go to slide go to slide

Book a Free Trial Class

Practice Questions on Converse of Pythagoras Theorem

Faqs on converse of pythagoras theorem.

The coverse of the Pythagoras theore m states that, in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

What is the Formula for Converse of Pythagoras Theorem?

The converse of Pythagoras theorem formula is c 2 = a 2 + b 2 , where a, b, and c are the sides of the triangle .

How Do You Prove the Converse of Pythagoras Theorem?

The converse of the Pythagoras theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If we consider two triangles ΔABC and ΔPQR where c 2 = a 2 + b 2 then we can say ∠C is a right triangle.

What is the Converse of the Pythagoras Theorem Useful For?

The converse of the Pythagoras theorem is useful in determining if a triangle is a right triangle or not. Whereas, a Pythagorean theorem helps in determining the length of the missing side of a right triangle.

What is the Application of the Converse of Pythagoras Theorem?

The application of the converse of the Pythagoras theorem is that the measurements help in determining what type of a triangle it is. There are three scenarios that we can determine, they are:

• If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle.
• If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle.
• If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle.

What is the Pythagorean Theorem?

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can say, (Opposite) 2 + (Adjacent) 2 = (Hypotenuse) 2 ​​​​​​.

• Math Article
• Converse Of Pythagoras Theorem

Converse of Pythagoras Theorem

The converse of Pythagoras theorem  states that “If the square of a side is equal to the sum of the square of the other two sides, then triangle must be right angle triangle”. Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is said that if this relation satisfies, then triangle must be right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy given condition a 2 + b 2 = c 2 , then the triangle is a right-angle triangle.

Let us see the proof of this theorem along with examples.

Converse of Pythagoras Theorem Proof

Statement: If the length of a triangle is a, b and c and c 2 = a 2 + b 2 , then the triangle is a right-angle triangle.

Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.

In △EGF, by Pythagoras Theorem:

EF 2 = EG 2 + FG 2 = b 2 + a 2 …………(1)

In △ABC, by Pythagoras Theorem:

AB 2 = AC 2 + BC 2 = b 2 + a 2 …………(2)

From equation (1) and (2), we have;

EF 2 = AB 2

⇒ △ ACB ≅ △EGF (By SSS postulate)

⇒ ∠G is right angle

Thus, △EGF is a right triangle.

Hence, we can say that the converse of Pythagorean theorem also holds.

Hence Proved.

As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:

Where a, b and c are the sides of a triangle.

Applications

Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.

Converse of Pythagoras Theorem Examples

Question 1: The sides of a triangle are 5, 12 and 13. Check whether the given triangle is a right triangle or not?

Solution: Given,

By using the converse of Pythagorean Theorem,

a 2 +b 2 = c 2

c 2 = a 2 +b 2

Substitute the given values in the above equation,

13 2 = 5 2 +12 2

169 = 25 + 144

So, the given lengths are does not satisfy the above condition.

Therefore, the given triangle is a right triangle.

Question 2: The sides of a triangle are 7, 11 and 13. Check whether the given triangle is a right triangle or not?

Solution: Given;

Substitute the given values in the the above equation,

13 2 = 7 2 + 11 2

169 = 49 + 121

So, it is not satisfied with the above condition.

Therefore, the given triangle is not a right triangle.

Question 3: The sides of a triangle are 4,6 and 8. Say whether the given triangle is a right triangle or not.

Solution: Given: a = 4, b = 6, c = 8

By the converse of Pythagoras theorem

8 2 = 4 2 + 6 2

64 = 16 + 36

The sides of the given triangle do not satisfy the condition a 2 +b 2 = c 2 .

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Search Functionality Update!

To optimize your search experience, please refresh the page.

Windows: Press Ctrl + F5

Mac: Use Command + Option + R

Mobile: Tap and hold the refresh icon, then select "Hard Refresh" or "Reload Without Cache" for an instant upgrade!

• Child Login
• Number Sense
• Measurement
• Pre Algebra
• Figurative Language
• Reading Comprehension
• Reading and Writing
• Science Worksheets
• Social Studies Worksheets
• Math Worksheets
• ELA Worksheets
• Online Worksheets

Browse By Grade

• Become a Member

• Kindergarten

• Skip Counting
• Place Value
• Number Lines
• Subtraction
• Multiplication
• Word Problems
• Comparing Numbers
• Ordering Numbers
• Odd and Even Numbers
• Prime and Composite Numbers
• Roman Numerals
• Ordinal Numbers

• Big vs Small
• Long vs Short
• Tall vs Short
• Heavy vs Light
• Full and Empty
• Metric Unit Conversion
• Customary Unit Conversion
• Temperature

• Tally Marks
• Mean, Median, Mode, Range
• Mean Absolute Deviation
• Stem and Leaf Plot
• Box and Whisker Plot
• Permutations
• Combinations

• Lines, Rays, and Line Segments
• Points, Lines, and Planes
• Transformation
• Ordered Pairs
• Midpoint Formula
• Distance Formula
• Parallel and Perpendicular Lines
• Surface Area
• Pythagorean Theorem

• Significant Figures
• Proportions
• Direct and Inverse Variation
• Order of Operations
• Scientific Notation
• Absolute Value

• Translating Algebraic Phrases
• Simplifying Algebraic Expressions
• Evaluating Algebraic Expressions
• Systems of Equations
• Slope of a Line
• Equation of a Line
• Quadratic Equations
• Polynomials
• Inequalities
• Determinants
• Arithmetic Sequence
• Arithmetic Series
• Geometric Sequence
• Complex Numbers
• Trigonometry

Converse of the Pythagorean Theorem Worksheets

• Geometry >
• Pythagorean Theorem >

Give children access to our pdf converse of the Pythagorean theorem worksheets, so they develop proficiency in finding whether a triangle, with the three side dimensions given, is a right triangle or not. Guide the enthusiastic minds to apply the Pythagorean theorem and determine whether the given triangle is a right triangle. Each printable worksheet consists of four illustrated exercises and two word-format questions, where students verify if the square of the largest number is equal to the sum of the two other numbers. Our "Is It a Right Triangle Worksheets", with included answer key, are available in both customary and metric units.

These printable resources are suitable for 7th grade, 8th grade, and high school students.

Related Printable Worksheets

▶ Pythagorean Theorem Charts

▶ Finding the Hypotenuse

▶ Finding the Unknown Side Lengths in Right Triangles

Tutoringhour

What we offer, information.

• Membership Benefits
• How to Use Online Worksheets
• How to Use Printable Worksheets
• Printing Help
• Testimonial
• Privacy Policy
• Refund Policy

Copyright © 2024 - Tutoringhour

You must be a member to unlock this feature!

Sign up now for only $29.95/year — that's just 8 cents a day!

Printable Worksheets

• 20,000+ Worksheets Across All Subjects
• Access to Answer Key
• Add Worksheets to "My Collections"
• Create Custom Workbooks

Digitally Fillable Worksheets

• 1100+ Math and ELA Worksheets
• Preview and Assign Worksheets
• Create Groups and Add Children
• Track Progress

Pythagorean Converse & Inequalities Color-by-Number

Also included in

Description

Questions & answers, acute math store.

• We're hiring
• Help & FAQ
• Privacy policy
• Student privacy
• Terms of service
• Tell us what you think

• For Teachers
• Schools & Districts

• Enter code ••• Enter Code

Free Online converse pythagoras theorem Flashcards

Master the Pythagoras Theorem with our curated collection of Converse Pythagoras Theorem flashcards on Quizizz. Learn, practice, and excel in math!

Explore Flashcards by Subject

• Number Sense
• Math Word Problems
• probability and statistics
• Trigonometry
• Measurement
• Multiplication
• Mixed Operations
• Data and Graphing
• Subtraction
• arithmetic and number theory
• Percents, Ratios, and Rates
• Math Puzzles
• Expressions and Equations
• Linear Functions
• Inequalities and System of Equations
• Monomials Operations
• Polynomial Operations
• Radical Expressions
• Rational Expressions
• Statistics and Probabilities
• Fundamentals and Building Blocks
• Equations and Inequalities
• System of Equations and Quadratic
• Complex Numbers
• Functions Operations
• Conic Sections
• Trigonometric Functions
• Sequences and Series
• Graphs & Functions
• Probability & Combinatorics

Explore Flashcards by Grades

• Kindergarten

Explore Free Printable converse pythagoras theorem Flashcard

Explore the world of geometry with our Converse Pythagoras Theorem flashcards on Quizizz. These flashcards are designed to help students understand and memorize the theorem in an interactive and engaging manner. They provide a visual demonstration of the theorem, thereby making it easier for students to comprehend. The flashcards are not only informative but also challenging, keeping students intrigued while learning. Whether you're a beginner trying to grasp the basics or an advanced learner seeking to perfect your knowledge, these flashcards are an excellent resource for you. Quizizz offers a versatile platform that is favored by teachers for its ease of use and flexibility of game modes. It provides a comprehensive library of resources, including our Converse Pythagoras Theorem flashcards, enabling teachers to create tailored quizzes for unit reviews, test preparation, and direct instruction. The platform also includes AI features and various question types, making it more educational than other tools. Teachers can monitor individual student progress and navigate resources with ease. The free features and report functions of Quizizz make it a favorite educational tool among educators.