Problem 61
Question
Geometry Write a function that gives the length of the hypotenuse of an isosceles right triangle with side length \(s\) . Evaluate the inverse of the function to find the side length of an isosceles right triangle with a hypotenuse of 6 in.
Step-by-Step Solution
Verified Answer
The length of the hypotenuse of an isosceles right triangle with side length 6 is \(6\sqrt{2}\) in. The side length of an isosceles right triangle with a hypotenuse of 6 in. is \(3\sqrt{2}\) in.
1Step 1: Define Function for Hypotenuse
The function to calculate the hypotenuse ('h') of an isosceles right triangle with side length ('s') can be defined as: \(h(s) = \sqrt{s^2 + s^2} = s\sqrt{2}\).
2Step 2: Evaluate Hypotenuse Function
By substituting the value of \(s = 6\) into the function \(h(s)\), we get: \(h(6) = 6\sqrt{2} in.\)
3Step 3: Define Inverse Function for Side Length
The inverse function to find the side length ('s') given the hypotenuse ('h') can be defined as: \(s(h) = \frac{h}{\sqrt{2}}\).
4Step 4: Evaluate Inverse Function
By substituting \(h = 6 in.\) into the function \(s(h)\), we get: \(s(6) = \frac{6}{\sqrt{2}} = 3\sqrt{2} in. \)
Key Concepts
Isosceles Right TriangleHypotenuseInverse FunctionSide Length
Isosceles Right Triangle
An isosceles right triangle is a special type of triangle that has two sides of equal length and a right angle between them. This means that two angles of the triangle are also equal, each measuring 45 degrees. The characteristics of an isosceles right triangle make it a straightforward subject in geometry due to its symmetrical properties.
Understanding these triangles involves recognizing how their sides and angles relate. Since two angles are equal and the sum of angles in any triangle is 180 degrees, it's easy to deduce that if two angles are 45 degrees, the third must be 90 degrees. This provides the defining right angle of the isosceles right triangle.
When working with these triangles, the equal sides are referred to as the legs, and the side opposite the right angle is known as the hypotenuse. This relationship among the sides and angles is crucial for calculations involving trigonometry and geometry.
Understanding these triangles involves recognizing how their sides and angles relate. Since two angles are equal and the sum of angles in any triangle is 180 degrees, it's easy to deduce that if two angles are 45 degrees, the third must be 90 degrees. This provides the defining right angle of the isosceles right triangle.
When working with these triangles, the equal sides are referred to as the legs, and the side opposite the right angle is known as the hypotenuse. This relationship among the sides and angles is crucial for calculations involving trigonometry and geometry.
Hypotenuse
The hypotenuse is the longest side of a right triangle, found opposite the right angle. In an isosceles right triangle, the hypotenuse can be calculated using the Pythagorean theorem. This relationship is expressed as: if each leg (or side length, 's') is equal, then the length of the hypotenuse, 'h', can be calculated using the formula:
\[ h = s\sqrt{2} \]
This formula arises from the Pythagorean theorem, where \(a^2 + b^2 = c^2\), and here both \(a\) and \(b\) are equal to \(s\). Hence, we have \(s^2 + s^2 = h^2\), which simplifies to \(h = s\sqrt{2}\).
This relationship is fundamental in solving problems related to isosceles right triangles, as determining the hypotenuse is often a step in understanding the shape's dimensions.
\[ h = s\sqrt{2} \]
This formula arises from the Pythagorean theorem, where \(a^2 + b^2 = c^2\), and here both \(a\) and \(b\) are equal to \(s\). Hence, we have \(s^2 + s^2 = h^2\), which simplifies to \(h = s\sqrt{2}\).
This relationship is fundamental in solving problems related to isosceles right triangles, as determining the hypotenuse is often a step in understanding the shape's dimensions.
Inverse Function
Inverse functions allow us to reverse processes in mathematics, making it possible to find original values given results. For example, when given the hypotenuse of an isosceles right triangle, you can determine the side length using an inverse function.
The inverse function for finding the side length \(s\) of an isosceles right triangle when the hypotenuse \(h\) is known is defined as follows:
\[ s(h) = \frac{h}{\sqrt{2}} \]
This function essentially "undoes" the hypotenuse formula \(h = s\sqrt{2}\) by isolating \(s\). In applying inverse functions, remember you need to rearrange the equation to solve for the variable of interest. Using this inverse function is particularly helpful when solving exercises where you begin with a known hypotenuse and seek to uncover the original side lengths.
The inverse function for finding the side length \(s\) of an isosceles right triangle when the hypotenuse \(h\) is known is defined as follows:
\[ s(h) = \frac{h}{\sqrt{2}} \]
This function essentially "undoes" the hypotenuse formula \(h = s\sqrt{2}\) by isolating \(s\). In applying inverse functions, remember you need to rearrange the equation to solve for the variable of interest. Using this inverse function is particularly helpful when solving exercises where you begin with a known hypotenuse and seek to uncover the original side lengths.
Side Length
The side length in an isosceles right triangle is a significant measurement, as it is one of the two equal legs of the triangle. Calculating the side length becomes straightforward once you know the length of the hypotenuse. Using the inverse function, we can determine the side length \(s\) from the given hypotenuse \(h\) as:
\[ s = \frac{h}{\sqrt{2}} \]
When tasked with determining the side of an isosceles right triangle, if you start with the hypotenuse, you'll use this formula to "work backwards" and find \(s\).
This function is often required in exercises that involve geometric relationships or transformations, where knowing the precise dimensions of these simple yet elegant triangles helps accomplish more complex algebraic or geometric tasks.
\[ s = \frac{h}{\sqrt{2}} \]
When tasked with determining the side of an isosceles right triangle, if you start with the hypotenuse, you'll use this formula to "work backwards" and find \(s\).
This function is often required in exercises that involve geometric relationships or transformations, where knowing the precise dimensions of these simple yet elegant triangles helps accomplish more complex algebraic or geometric tasks.
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