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Magazine Chapter 9: Problem 3
In a right triangle with legs a and b, if the lengths of the legs are doubled, then the length of the hypotenuse is
Short Answer
Expert verified
The length of the new hypotenuse is 2c.
Step by step solution
01
Understand the Problem
A right triangle has legs of lengths a and b. If these lengths are doubled, we need to find the length of the new hypotenuse.
02
Original Hypotenuse
First, recall the Pythagorean theorem which states that in a right triangle, the length of the hypotenuse c is given by: \[c = \sqrt{a^2 + b^2}\]
03
Doubling the Lengths
When the lengths of the legs a and b are doubled, the new lengths will be 2a and 2b.
04
New Hypotenuse Calculation
We now apply the Pythagorean theorem to find the new hypotenuse c': \[c' = \sqrt{(2a)^2 + (2b)^2}\]
05
Simplify the Expression
Simplify the expression inside the square root: \[c' = \sqrt{4a^2 + 4b^2} = \sqrt{4(a^2 + b^2)}\]
06
Factor Out the 4
Factor out the 4 from the square root: \[c' = \sqrt{4} \times \sqrt{a^2 + b^2}\]
07
Complete the Calculation
\[\sqrt{4} = 2\] and we know \[\sqrt{a^2 + b^2} = c\] (the original hypotenuse) so,\[c' = 2c\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Right Triangles
A right triangle is a type of triangle that has one angle measuring 90 degrees. This angle is called the right angle. The sides forming this right angle are known as the legs of the triangle, which we generally call 'a' and 'b'.
The longest side of the right triangle, opposite the right angle, is called the hypotenuse, often labeled as 'c'. Due to the right angle, the legs and the hypotenuse follow a special relationship defined by the Pythagorean theorem.
Understanding how right triangles behave is key to solving problems related to their dimensions.
The longest side of the right triangle, opposite the right angle, is called the hypotenuse, often labeled as 'c'. Due to the right angle, the legs and the hypotenuse follow a special relationship defined by the Pythagorean theorem.
Understanding how right triangles behave is key to solving problems related to their dimensions.
The Hypotenuse
In a right triangle, the hypotenuse is the side opposite the right angle and is the longest side. The Pythagorean theorem helps us calculate the hypotenuse when the lengths of the legs are known.
According to the Pythagorean theorem, the hypotenuse 'c' can be found using the formula: \[ c = \sqrt{a^2 + b^2} \]
This formula connects the lengths of the legs to the hypotenuse, providing a clear pathway for understanding the relationships within the triangle.
According to the Pythagorean theorem, the hypotenuse 'c' can be found using the formula: \[ c = \sqrt{a^2 + b^2} \]
This formula connects the lengths of the legs to the hypotenuse, providing a clear pathway for understanding the relationships within the triangle.
Algebraic Simplification
Simplifying algebraic expressions is essential for solving mathematical problems efficiently. When dealing with the Pythagorean theorem and any modifications to the lengths of the legs, simplification comes into play.
For instance, if the legs of the triangle are doubled, we replace 'a' with '2a' and 'b' with '2b' in the theorem: \[ c' = \sqrt{(2a)^2 + (2b)^2} = \sqrt{4a^2 + 4b^2} = \sqrt{4(a^2 + b^2)} \] The expression within the square root is simplified by factoring out the common term: \[ \sqrt{4(a^2 + b^2)} = 2 \sqrt{a^2 + b^2} = 2c \] This use of algebraic simplification makes the problem more manageable and leads to a clear solution.
For instance, if the legs of the triangle are doubled, we replace 'a' with '2a' and 'b' with '2b' in the theorem: \[ c' = \sqrt{(2a)^2 + (2b)^2} = \sqrt{4a^2 + 4b^2} = \sqrt{4(a^2 + b^2)} \] The expression within the square root is simplified by factoring out the common term: \[ \sqrt{4(a^2 + b^2)} = 2 \sqrt{a^2 + b^2} = 2c \] This use of algebraic simplification makes the problem more manageable and leads to a clear solution.
Mathematical Reasoning
Mathematical reasoning is the process of using logical thinking to solve problems and understand concepts. When we apply the Pythagorean theorem and other algebraic principles, we use reasoning to break down and solve the problem step by step.
Take the problem of finding the new hypotenuse when the legs of the triangle are doubled. We start with the original theorem, adapt it to the new lengths, simplify the expression, and conclude with the final equation: \[ c' = 2c \] This process involves reasoning through each step, ensuring the logical flow from the starting point to the conclusion. Reasoning helps validate each transformation and simplification, making sure our results are correct and meaningful.
Take the problem of finding the new hypotenuse when the legs of the triangle are doubled. We start with the original theorem, adapt it to the new lengths, simplify the expression, and conclude with the final equation: \[ c' = 2c \] This process involves reasoning through each step, ensuring the logical flow from the starting point to the conclusion. Reasoning helps validate each transformation and simplification, making sure our results are correct and meaningful.
