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Chapter 2: Problem 7

Let \(P\) be the set of right triangles with a \(5^{\prime \prime}\) hypotenuse and whose height and length are \(a\) and \(b\), respectively. Characterize the outcomes in \(P\).

Short Answer

Expert verified
The possible outcomes in the given set P are characterized by dimensions of \(a\) and \(b\) of a right triangle such that \(0 \leq a \leq 5\) and \(b = \sqrt{25 - a^2}\).

Step by step solution

01

Apply Pythagorean Theorem

According to the Pythagorean theorem, \(a^2 + b^2 = 5^2\). Solve this equation for one variable to express it in terms of the other. Let's express \(b\) in terms of \(a\). We can rearrange the equation to look like this: \(b^2 = 5^2 - a^2\).
02

Solve for b

To isolate \(b\), we take the square root of both sides of the equation: \(b = \sqrt{5^2 - a^2}.\)
03

Characterize the outcomes

Since \(a\) and \(b\) denote the lengths of the sides of a triangle, they must be positive, so \(0 \leq a \leq 5\), and similarly for \(b\). Therefore, the outcomes will be characterized by values of \(a\) and \(b\) such that \(0 \leq a \leq 5\) and \(b = \sqrt{5^2 - a^2}\)

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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 exactly 90 degrees. This makes it unique compared to other triangles. The right-angle is an important feature as it allows us to use the Pythagorean Theorem to find the lengths of the sides.
  • One angle is 90 degrees.
  • Other two angles are complementary and less than 90 degrees each.
  • The sides have special names: the two sides that form the right angle are called 'legs,' and the side opposite the right angle is known as the 'hypotenuse.'
Right triangles are fundamental in geometry and trigonometry. They provide the basis for many practical applications like navigation and construction.
Their properties are also widely used in solving real-world problems, making them essential for students to understand.
The Hypotenuse Explained
The hypotenuse is the longest side of a right triangle. It lies opposite the right angle. This makes it simple to identify in any right triangle diagram, and it is the most crucial component when applying the Pythagorean Theorem.
  • Always opposite the 90-degree angle.
  • Usually represented by the letter 'c' in equations.
  • Can be found if the other two sides are known using the formula: \(c = \sqrt{a^2 + b^2}\)
In the given problem, the hypotenuse is known to be 5 inches. This information allows us to solve for the other sides of the triangle using the Pythagorean Theorem. The consistent measure makes it easier to characterize outcomes and explore other properties of the triangle.
Understanding Triangle Sides in Pythagorean Theorem
The sides of a right triangle, beside the hypotenuse, are essential for using the Pythagorean Theorem. These two sides are often referred to as 'legs' of the triangle, typically marked as 'a' and 'b.' They form the right angle.
  • They work together according to: \(a^2 + b^2 = c^2\)
  • The legs can vary in length, but their combined squared values must always equate to the square of the hypotenuse.
  • For practical evaluation, rearranging the equation can help to solve for unknown side lengths like \(b = \sqrt{c^2 - a^2}\).
In the original problem, given that \(c = 5\), the focus was on expressing one leg in terms of the other using the equation \(b = \sqrt{5^2 - a^2}\). This approach helps us find all possible combinations for 'a' and 'b' that satisfy the triangle's properties when the hypotenuse is fixed at 5 inches. It shows how important each side's relationship is when working with right triangles.

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Most popular questions from this chapter

A toy manufacturer buys ball bearings from three different suppliers \(-50 \%\) of her total order comes from supplier \(1,30 \%\) from supplier 2, and the rest from supplier \(3 .\) Past experience has shown that the quality-control standards of the three suppliers are not all the same. Two percent of the ball bearings produced by supplier 1 are defective, while suppliers 2 and 3 produce defective bearings \(3 \%\) and \(4 \%\) of the time, respectively. What proportion of the ball bearings in the toy manufacturer's inventory are defective?

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