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Perpendiculars and Distance Page 215 You proved that two lines are parallel using angle relationships. • Find the distance between a point and a line. • Find the distance between parallel lines. Page 215 Construct Distance From Point to a Line CONSTRUCTION A certain roof truss is designed so that the center post extends from the peak of the roof (point A) to the main beam. Construct and name the segment whose length represents the shortest length of wood that will be needed to connect the peak of the roof to the main beam. The distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point. Locate points R and S on the main beam equidistant from point A. Locate a second point not on the beam equidistant from R and S. Construct AB so that AB is perpendicular to the beam. Answer: The measure of 𝐴𝐵 represents the shortest length of wood needed to connect the peak of the roof to the main beam. KITES Which segment represents the shortest distance from point A to DB? A. AD B. AB C. CX D. AX COORDINATE GEOMETRY Line s contains points at (0, 0) and (–5, 5). Find the distance between line s and point V(1, 5). Step 1 Find the slope of line s. (–5, 5) V(1, 5) (0, 0) Begin by finding the slope of the line through points (0, 0) and (–5, 5). Then write the equation of this line by using the point (0, 0) on the line. Slope-intercept form m = –1, (x1, y1) = (0, 0) Simplify. The equation of line s is y = –x. Step 2 Write an equation of the line t perpendicular to line s through V(1, 5). Since the slope of line s is –1, the slope of line t is 1. Write the equation for line t through V(1, 5) with a slope of 1. Slope-intercept form m = 1, (x1, y1) = (1, 5) Simplify. Subtract 1 from each side. The equation of line t is y = x + 4. Step 3 line s: Solve the system of equations to determine the point of intersection. y = –x line t: (+) y = x + 4 2y = 4 Add the two equations. y= 2 Divide each side by 2. Solve for x. 2 = –x –2 = x Substitute 2 for y in the first equation. Divide each side by –1. The point of intersection is (–2, 2). Let this point be Z. Step 4 Use the Distance Formula to determine the distance between Z(–2, 2) and V(1, 5). Distance formula Substitution Simplify. Answer: The distance between the point and the line is or about 4.24 units. Page 218 By definition, parallel lines do not intersect. An alternate definition states that two lines in a plane are parallel if they are everywhere equidistant. Equidistant means that the distance between two lines measured along a perpendicular line to the lines is always the same. Steps to follow… 1. Find the slope 2. Write the equation of the line. 3. Write the equation of a line perpendicular through the point given. 4. Take equations from #2 and #3 and solve for x and y using elimination. 5. Use the distance formula to find the distance between the given point and the point you found in #4. 𝑑 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2 Page 218 Find the distance between the parallel lines a and b whose equations are y = 2x + 3 and y = 2x – 1, respectively. You will need to solve a system a b of equations to find the endpoints of a segment that is perpendicular to both a and b. From their equations, we know that the slope of line a and line b is 2. p Sketch line p through the y-intercept of line b, (0, –1), perpendicular to lines a and b. Write an equation for line p. The slope of p is the opposite reciprocal of Use the y-intercept of line b, (0, –1), as one of the endpoints of the perpendicular segment. Point-slope form Simplify. Subtract 1 from each side. Use a system of equations to determine the point of intersection of the lines a and p. Substitute 2x + 3 for y in the second equation. Group like terms on each side. Simplify on each side. Multiply each side by Substitute equation for p. . for x in the Simplify. Step 3 Use the Distance Formula to determine the distance between (0, –1) and (–1.6, –0.2). Distance Formula x2 = –1.6, x1 = 0, y2 = –0.2, y1 = –1 Answer: The distance between the lines is about 1.79 units. Steps to follow… 1. Find the slope 2. Write the equation of the line. 3. Write the equation of a line perpendicular through the point given. 4. Take equations from #2 and #3 and solve for x and y using elimination. 5. Use the distance formula to find the distance between the given point and the point you found in #4. 𝒅 = 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐 3-6 Assignment Page 221, 13, 14, 15, 17, 21, 23