Question

On the map of Milwaukee, Wisconsin, the shape formed by Route 190 , Route 57, and Route 145 closely resembles an isosceles right triangle. If the distances on Routes 190 and 57 are each \(2.8\) miles, find the distance on Route 145 between Route 190 and Route 57 . Round to the nearest tenth.

Short answer

The distance on Route 145 is approximately 4.0 miles.

Step-by-step solution

Understanding the Problem

The problem states that Route 190, Route 57, and Route 145 form an isosceles right triangle on the map. An isosceles right triangle has two sides that are of equal length and a hypotenuse. In this exercise, the two sides are each 2.8 miles.

Applying the Pythagorean Theorem

To find the hypotenuse (Route 145), we can use the Pythagorean theorem which states: \[ c^2 = a^2 + b^2 \]where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

Substituting Known Values

For this problem, substitute the lengths of the two equal sides into the formula. So, we have:\[ c^2 = 2.8^2 + 2.8^2 \]

Calculating Squared Values

Calculate each square:\[ 2.8^2 = 7.84 \] Thus: \[ c^2 = 7.84 + 7.84 \]

Finding the Sum

Add the squared values:\[ c^2 = 15.68 \]

Solving for the Hypotenuse

Take the square root of both sides to solve for \(c\):\[ c = \sqrt{15.68} \]

Rounding the Answer

Calculate the square root and round to the nearest tenth:\[ c \approx 3.96 \] Rounded to the nearest tenth, \( c \approx 4.0 \).

Key concepts

Isosceles Right Triangle

An isosceles right triangle is a special type of triangle where two of its sides are of equal length, and it includes a right angle (90 degrees). This unique triangle is characterized by its symmetry and specific angle measures. The two equal sides are called the legs, while the longest side opposite the right angle is known as the hypotenuse. In our example, Routes 190 and 57 each have a length of 2.8 miles, forming the legs of the triangle. Understanding the features of an isosceles right triangle helps in recognizing that the two legs are congruent, and this is critical when applying mathematical concepts like the Pythagorean Theorem.

• Two equal legs: Identical in length, forming the base and height.
• Right angle: Confirms the presence of the hypotenuse.
• Symmetry: Often simplifies calculations and problem-solving.

Hypotenuse Calculation

The hypotenuse in a right triangle is the side opposite the right angle and is the longest side of the triangle. To find the hypotenuse of an isosceles right triangle, such as the one formed by the Milwaukee routes, we use the Pythagorean Theorem. This fundamental theorem in geometry states that in a right triangle, the square of the hypotenuse (\(c\)) equals the sum of the squares of the other two sides (\(a\) and \(b\)): \[c^2 = a^2 + b^2\]For our triangle, the legs are each 2.8 miles, hence:\[c^2 = 2.8^2 + 2.8^2\]The calculations involve squaring each leg and adding these values together. This provides the square of the hypotenuse, from which we take the square root to solve for \(c\), resulting in:\[c = \sqrt{15.68}\] Knowing how to determine the hypotenuse is essential in navigation and land survey projects, where such distances are often needed.

Rounding to Nearest Tenth

Rounding numbers is a mathematical technique used to simplify numbers, making them easier to work with. In our example, the distance of the hypotenuse was determined to be approximately 3.96 miles. To make this number simpler in terms of practical usage, we round to the nearest tenth. Here’s how you round to the nearest tenth:

• Identify the tenths digit (the first digit to the right of the decimal point).
• Look at the next digit (hundredths place) to determine if you round up or down.
• If this digit is 5 or more, round the tenths digit up by one.
• If it's less than 5, keep the tenths digit as is.

In the case of our hypotenuse calculation, 3.96 rounds up to 4.0 because the hundredths digit (6) is greater than 5. Rounding helps in presenting numbers sensibly, especially in contexts like construction or navigation where precision is relevant.