We start by reviewing the method for calculating the length...

We start by reviewing the method for calculating the length of a straight line segment. a. Find the length of the straight line segment from $(1,2)$ to $(5,4)$, shown in Figure 2. The lengths of the two sides of the right triangle are $4-2=2$ and $5-1=4$. You can use the Pythagorean Theorem to determine the length of the hypotenuse. b. Now generalize this example to find a formula for the length of the segment of the straight line $y=m x+c$ from $x=a$ to $x=b$. This is the line segment connecting the points ( $a$, $\qquad$ ) and (b, $\qquad$ ). Show the necessary work to get the length equal to $(b-a) \sqrt{1+m^2}$. Check your answer to part a using this formula. c. Write an expression for the length of the curve in Figure 3 made up of four straight line segments with slopes $m_1, m_2, m_3$ and $m_4$.

We start by reviewing the method for calculating the length of a straight line segment.
a. Find the length of the straight line segment from $(1,2)$ to $(5,4)$, shown in Figure 2. The lengths of the two sides of the right triangle are $4-2=2$ and $5-1=4$. You can use the Pythagorean Theorem to determine the length of the hypotenuse.
b. Now generalize this example to find a formula for the length of the segment of the straight line $y=m x+c$ from $x=a$ to $x=b$. This is the line segment connecting the points ( $a$, $\qquad$ ) and (b, $\qquad$ ). Show the necessary work to get the length equal to $(b-a) \sqrt{1+m^2}$. Check your answer to part a using this formula.
c. Write an expression for the length of the curve in Figure 3 made up of four straight line segments with slopes $m_1, m_2, m_3$ and $m_4$.

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Key Concepts

Distance Formula

The distance formula is a key concept used to determine the length between two points in the Cartesian plane. It is derived from the Pythagorean theorem and provides a method to compute the straight-line distance by considering the horizontal and vertical differences between the points.

Pythagorean Theorem

The Pythagorean theorem is fundamental in relating the sides of a right triangle. When a line segment is represented as the hypotenuse of a right triangle with horizontal and vertical legs, this theorem allows us to calculate its length, forming the basis for the distance formula.

Length of a Straight Line Segment

Calculating the length of a straight line segment involves using the differences in the coordinates of its endpoints to determine the lengths of the legs of a right triangle. This calculation can be generalized to any linear function in the form y = mx + c, yielding a formula of the form (b - a)?(1 + m²) when evaluating the segment between two specified x-values.

Piecewise Linear Curves

A piecewise linear curve is composed of multiple straight line segments connected end-to-end. The total length of such a curve is determined by summing the lengths of its individual segments, each of which can be computed using the distance formula or the generalized formula for a straight line segment.

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