Math Problem Statement
Solution
Let's break down and solve the problems shown in the image.
Problem 1: Find a Pythagorean triplet in which one member is 12.
A Pythagorean triplet consists of three integers , , and that satisfy the equation: When one of the members is 12, we can use the formula for generating triplets: where are positive integers. Let's substitute values to find a triplet containing 12.
For and : So the triplet is .
Now, we can scale this triplet by multiplying by a factor. To get 12, multiply the entire triplet by 3: Thus, the required triplet is .
Problem 2: Find the smallest number by which 256 must be multiplied to obtain a perfect cube.
To find the smallest number that makes 256 a perfect cube, we start by performing its prime factorization: For a number to be a perfect cube, the exponents in its prime factorization must be multiples of 3. Currently, has an exponent of 8, which is not a multiple of 3. The smallest multiple of 3 greater than 8 is 9.
Thus, we need to multiply 256 by (since ) to make the exponent of 2 equal to 9. Therefore, the smallest number by which 256 must be multiplied is:
Problem 3: The adjoining figure RUNS is a parallelogram. Find and (lengths in cms).
Given the diagram of the parallelogram RUNS:
- We are tasked with finding the values of and .
Using properties of parallelograms:
-
Opposite sides are equal. Therefore, , so we have: This does not provide new information.
-
For the diagonals, in a parallelogram, the diagonals bisect each other. This implies: Substituting the given expressions: Solving for : Now substitute into : Solving for :
Thus, the values of and are:
Would you like more details or explanations on any of these? Here are 5 related questions to expand your understanding:
- How can we generate other Pythagorean triplets using different values of and ?
- Why do we focus on prime factorization when finding the number that makes a given number a perfect cube?
- What are the properties of parallelograms that help in solving geometric problems like this?
- How can we solve for variables when given relationships between the diagonals in a parallelogram?
- What are other real-life applications of Pythagorean triplets?
Tip: In geometric problems, always remember the key properties of the shapes involved (e.g., bisecting diagonals in parallelograms) to simplify the solution!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Pythagorean triplet
Prime factorization
Geometric properties of parallelograms
Formulas
a^2 + b^2 = c^2
Prime factorization: 256 = 2^8
Diagonal bisection property of parallelograms: UM = UN
Theorems
Pythagoras' theorem
Properties of perfect cubes
Properties of parallelograms
Suitable Grade Level
Grades 7-9
Related Recommendation
Find Smallest Multiplier for 256 and Solve for x and y in a Parallelogram
Finding Squares and Pythagorean Triplets for Specific Numbers
Find Pythagorean Triplet (a, b, c) for Condition b + c = a^2
Geometry and Digit Problems: Solving Rectangle, Right Triangle, and Number Theory Challenges
Finding a Pythagorean Triplet (a, b, c) with Conditions a < b < c and b + c = a^2