Problem 21 - Prealgebra

Question

DARTS Darnell's first dart lands 2 inches to the right and 7 inches below the bull's-eye. What is the distance between the bull's-eye and where his first shot hit the target? Round to the nearest tenth of an inch.

Step-by-Step Solution

Answer

Darnell's dart landed approximately 7.3 inches from the bull's-eye.

Step 1: Understand the Problem

We need to find the straight line distance from the target's bull's-eye to the point where Darnell's dart hit. The dart lands 2 inches right and 7 inches below the bull's-eye, making these displacements in a right triangle's legs.

Step 2: Use the Pythagorean Theorem

Darnell's dart forms a right triangle with the bull's-eye. Use the Pythagorean Theorem, which states \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse (distance from bull's-eye to hit point). Set \(a = 2\) inches and \(b = 7\) inches.

Step 3: Plug in Known Values

According to the theorem, calculate \(c\) using: \[c^2 = 2^2 + 7^2 = 4 + 49 = 53\]

Step 4: Solve for c

Find the square root of 53 to determine \(c\): \(c = \sqrt{53} \approx 7.28011\).

Step 5: Round to the Nearest Tenth

Round the distance \(7.28011\) to the nearest tenth, resulting in \(c \approx 7.3\) inches.

Key Concepts

Right Triangles

A right triangle is one of the fundamental shapes in geometry. It is called "right" because it has one angle that measures exactly 90 degrees. This right angle is what makes calculations like measuring distances straightforward using the Pythagorean Theorem.
In any right triangle, the side opposite the right angle is called the "hypotenuse," and is always the longest side. The other two sides, which meet at the right angle, are known as the "legs." In many practical problems, such as the dartboard example above, these legs represent horizontal and vertical distances. Together, they form a characteristic L-shape.

• One key property of right triangles is that the sides satisfy the Pythagorean Theorem.
• This is useful for many tasks, including finding the shortest path between two points.

Whenever you spot a right triangle in a problem, remember that it could simplify how you approach an otherwise complex calculation.

Distance Calculation

Calculating distance between two points is often done using the straight line or "hypotenuse" of a right triangle, formed by the two points. This process can be simplified by using the Pythagorean Theorem.

• The theorem states: \(a^2 + b^2 = c^2\).
• Here, \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.

Consider the dartboard scenario: the dart lands 2 inches right and 7 inches below the bull's-eye. These displacements form the legs of a right triangle. By substituting these values into the Pythagorean Theorem, you get: \[c^2 = 2^2 + 7^2 = 4 + 49 = 53\]Ultimately, you solve for \(c\) by taking the square root, resulting in \(c = \sqrt{53}\). This method allows you to find the precise straight-line distance between any two points in a plane.

Rounding Numbers

Rounding is a math technique we use to simplify numbers, making them easier to handle and understand. In problems like calculating distances, exact values can sometimes be complex, so rounding helps communicate a close, simple estimate.
When rounding to the nearest tenth, you look at the number in the "hundredths" position. If this number is 5 or more, you round up the "tenths" digit by one. Otherwise, you leave the "tenths" digit as it is.
In the dartboard example, after calculating the distance, \(c = 7.28011\), the hundredths place is 8, which is more than 5. Therefore, the rounded distance becomes 7.3 inches.

• Rounding provides an easy-to-read approximation.
• It is often used when exact precision is unnecessary or when results need to be communicated clearly.

Remember, while rounding makes numbers simpler, it also means some loss of precision. Always consider how much accuracy is needed in your calculations.