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Chapter 1: Problem 13

Pythagorean Triples. Three natural numbers \(a, b,\) and \(c\) with \(a

Short Answer

Expert verified
(a) All the given triples satisfy the \(a^2 + b^2 = c^2\) equation, so they are Pythagorean triples. (b) Yes, there exists a Pythagorean triple of the form \(m, m+7,\) and \(m+8\), where \(m\) is a natural number. The Pythagorean triple for this case is (3, 10, 11). (c) No, there does not exist a Pythagorean triple of the form \(m, m+11,\) and \(m+12\), where \(m\) is a natural number.

Step by step solution

01

Verification of Pythagorean triples

To verify that each of the following triples of natural numbers forms a Pythagorean triple, we need to check if they satisfy the equation \(a^2 + b^2 = c^2\). Let's verify each separately: 1) 3, 4, and 5: \(3^2 + 4^2 = 9 + 16 = 25; 5^2 = 25 \Rightarrow 3^2 + 4^2 = 5^2\). 2) 6, 8, and 10: \(6^2 + 8^2 = 36 + 64 = 100; 10^2 = 100 \Rightarrow 6^2 + 8^2 = 10^2\). 3) 8, 15, and 17: \(8^2 + 15^2 = 64 + 225 = 289; 17^2 = 289 \Rightarrow 8^2 + 15^2 = 17^2\). 4) 10, 24, and 26: \(10^2 + 24^2 = 100 + 576 = 676; 26^2 = 676 \Rightarrow 10^2 + 24^2 = 26^2\). 5) 12, 35, and 37: \(12^2 + 35^2 = 144 + 1225 = 1369; 37^2 = 1369 \Rightarrow 12^2 + 35^2 = 37^2\). 6) 14, 48, and 50: \(14^2 + 48^2 = 196 + 2304 = 2500; 50^2 = 2500 \Rightarrow 14^2 + 48^2 = 50^2\). All of the given triples satisfy the \(a^2 + b^2 = c^2\) equation, so they are Pythagorean triples.
02

Checking the form \(m, m+7\), and \(m+8\)

Let's check if there exists a Pythagorean triple of the form \(m, m+7,\) and \(m+8\) for some natural number \(m\). We need to check if \(m^2 + (m+7)^2 = (m+8)^2\). Expanding the equation, we get: \[m^2 + m^2 + 14m + 49 = m^2 + 16m + 64\] Simplifying, we get: \[m^2 - 2m - 15 = 0\] Solving for m, we have: \[m(m-2) = 15\] Since m is a natural number, m must be a factor of 15. The factors of 15 are 1, 3, 5, and 15. Checking each, we find that only \(m = 3\) satisfies the equation, giving us the Pythagorean triple (3, 10, 11).
03

Checking the form \(m, m+11\), and \(m+12\)

Let's check if there exists a Pythagorean triple of the form \(m, m+11\), and \(m+12\) for some natural number \(m\). We need to check if \(m^2 + (m+11)^2 = (m+12)^2\). Expanding the equation, we get: \[m^2 + m^2 + 22m + 121 = m^2 + 24m + 144\] Simplifying, we get: \[m^2 - 2m - 23 = 0\] For the given equation, no natural number m satisfies this equation. We can prove this by factoring: \[m^2 - 2m - 23 = (m-5)(m+3)\] The left-hand side (LHS) of the equation is negative when \(m < 5\), and the right-hand side (RHS) is always positive. When \$m >= 5\$, the LHS becomes positive, while the RHS remains positive. Since the LHS is negative and the RHS is positive, no natural number m will satisfy the equation. Therefore, no such Pythagorean triple exists for the form \(m, m+11,\) and \(m+12\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry. It is used to relate the lengths of the sides in a right triangle. A right triangle is a triangle where one of the angles is exactly 90 degrees. The theorem states that in such a triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it is expressed as:
  • \( a^2 + b^2 = c^2 \)
Here, \( a \) and \( b \) are the lengths of the legs, and \( c \) is the hypotenuse. Pythagorean triples are specific sets of three natural numbers that fit this formula and describe the side lengths of a right triangle.
Natural Numbers
Natural numbers are a set of positive integers starting from 1, going onwards like 1, 2, 3, and so forth. They do not include zero or any negative numbers. In the context of the Pythagorean triples, the numbers \(a, b, \) and \(c\) are natural numbers. When these numbers satisfy the equation \(a^2 + b^2 = c^2\), they represent the side lengths of a right triangle. For example, the numbers 3, 4, and 5 form a well-known Pythagorean triple, because when squared and summed, \(3^2 + 4^2 = 5^2\). Each of these numbers is a natural number, important in verifying and exploring properties of geometric figures.
Right Triangle
A right triangle is a type of triangle with one of its angles measuring 90 degrees. In geometry, the Pythagorean Theorem applies specifically to right triangles. The two sides forming the right angle are called the legs, and the side opposite the right angle is the hypotenuse. For every right triangle, this relationship holds true:
  • The sum of the squares of the legs equals the square of the hypotenuse.
Finding Pythagorean triples involves determining sets of integers that can act as the sides of a right triangle. A commonly used example is the side lengths of 3, 4, and 5. Placing them within the theorem verifies that they satisfy \(3^2 + 4^2 = 5^2\), thus confirming the Pythagorean relationship in right triangles.
Verification of Equations
Verification of equations is a process used to confirm whether a set of numbers fulfills the Pythagorean Theorem. To verify given numbers as a Pythagorean triple, one must ensure that they satisfy \( a^2 + b^2 = c^2 \). In the exercise, you are provided with multiple sets of numbers, such as 3, 4, and 5, to check.
  • Start by squaring each of the numbers.
  • Then add the squares of the two smallest numbers.
  • Finally, check if this sum equals the square of the largest number.
If the equation holds true, then they form a Pythagorean triple. The verification process helps to solidify understanding of how the theorem is applied and ensures accuracy in identifying suitable triples.

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Most popular questions from this chapter

More Work with Pythagorean Triples. In Exercise (13), we verified that each of the following triples of natural numbers are Pythagorean triples: $$ \begin{array}{lll} \cdot 3,4, \text { and } 5 & \bullet 8,15, \text { and } 17 & \bullet 12,35, \text { and } 37 \\ \cdot 6,8, \text { and } 10 & \bullet 10,24, \text { and } 26 & \bullet 14,48, \text { and } 50 \end{array} $$ (a) Focus on the least even natural number in each of these Pythagorean triples. Let \(n\) be this even number and find \(m\) so that \(n=2 m .\) Now try to write formulas for the other two numbers in the Pythagorean triple in terms of \(m\). For example, for \(3,4,\) and \(5, n=4\) and \(m=2,\) and for 8\. \(15,\) and \(17, n=8\) and \(m=4 .\) Once you think you have formulas. test your results with \(m=10\). That is, check to see that you have a Pythagorean triple whose smallest even number is 20 . (b) Write a proposition and then write a proof of the proposition. The proposition should be in the form: If \(m\) is a natural number and \(m \geq 2\), then \(\ldots \ldots\)

Exploring Propositions. In Progress Check 1.2, we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of \(2\left(2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}, \ldots\right)\) and examine the units digits of these numbers, we could make the following conjectures (among others): \- If \(n\) is a natural number, then the units digit of \(2^{n}\) must be \(2,4,6,\) or 8 . \- The units digits of the successive powers of 2 repeat according to the pattern " 2,4,8,6 ." (a) Is it possible to formulate a conjecture about the units digits of successive powers of \(4\left(4^{1}, 4^{2}, 4^{3}, 4^{4}, 4^{5}, \ldots\right) ?\) If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form \(7^{n}-2^{n},\) where \(n\) is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form "If \(n\) is a natural number, then \(\ldots . "\) (c) Let \(f(x)=e^{2 x}\). Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, "If \(n\) is a natural number, then \(\ldots . "\)

Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, \(a, b,\) and \(c\) are real numbers. Theorem If \(f\) is a quadratic function of the form \(f(x)=a x^{2}+b x+c\) and \(a c<0,\) then the function \(f\) has two \(x\) -intercepts. Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g(x)=-8 x^{2}+5 x-2\) (b) \(h(x)=-\frac{1}{3} x^{2}+3 x\) (c) \(k(x)=8 x^{2}-5 x-7\) (d) \(j(x)=-\frac{71}{99} x^{2}+210\) (e) \(f(x)=-4 x^{2}-3 x+7\) (f) \(F(x)=-x^{4}+x^{3}+9\)

Construct a know-show table for each of the following statements and then write a formal proof for one of the statements. (a) If \(x\) is an even integer and \(y\) is an even integer, then \(x+y\) is an even integer. (b) If \(x\) is an even integer and \(y\) is an odd integer, then \(x+y\) is an odd integer. (c) If \(x\) is an odd integer and \(y\) is an odd integer, then \(x+y\) is an even integer.

Determine whether each of the following conditional statements is true or false. (a) If \(10<7,\) then \(3=4\). (c) If \(10<7,\) then \(3+5=8\). (b) If \(7<10,\) then \(3=4\). (d) If \(7<10,\) then \(3+5=8\).
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