Obj. 14 Perpendicular Lines
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Solve problems using perpendicular lines Set up and solve inequalities using the distance from a point to a line
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Write equations of a line parallel or perpendicular to a given line through a given point.
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Here are the steps to solve these problems:
1. Find the slopes of the two lines:
m1 = (8-2)/(5--2) = 6/3 = 2 (slope of r)
m2 = (7-0)/(-8--2) = 7/-6 = -1 (slope of s)
The slopes are negative reciprocals, so r ⊥ s.
2. The slopes are m1 = 2 and m2 = -1/2. Since m1 × m2 = -1, the lines are perpendicular.
3. The given line has slope 3. The perpendicular line will have slope -1/3. Plug into the point-slope form
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2. It defines a normal line as perpendicular to the tangent line and gives the condition for perpendicular lines as having slopes that are negative reciprocals of each other.
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The student is able to identify scale factors and use them to solve problems involving similar polygons. Scale factors allow comparison of corresponding side lengths of similar polygons through proportions. To determine if two polygons are similar, their corresponding angles must be congruent and side lengths must be proportional as shown through equal ratios.
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The document presents information about straight lines and coordinates systems. It discusses key concepts such as the Cartesian coordinate system developed by Rene Descartes, which specifies each point in a plane using perpendicular x and y axes. The document also covers the definition of a straight line as the shortest distance between two points, midpoint and centroid formulas, polar coordinates, slope as a measure of a line's steepness, and the general equation of a straight line. The presentation is delivered to a lecturer by five students who each discuss different aspects of straight lines and coordinate systems.
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Use slopes to identify parallel and perpendicular lines
Write equations of a line parallel or perpendicular to a given line through a given point.
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- The side-splitter theorem states that a line parallel to one side of a triangle divides the other two sides proportionally.
- The triangle proportionality theorem can be used to verify if two segments are parallel.
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Here are the steps to solve these problems:
1. Find the slopes of the two lines:
m1 = (8-2)/(5--2) = 6/3 = 2 (slope of r)
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Obj. 14 Perpendicular Lines
- 1. Obj. 14 Perpendicular Lines The student will be able to (I can): • Solve problems using perpendicular lines • Set up and solve inequalities using the distance from a point to a line
- 2. Perpendicular Transversal Theorem If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. ∠1 ≅ ∠2 ⇒ m ⊥ n If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. n p & n ⊥q ⇒ p ⊥ q 1 2m n n p q
- 3. Distance From a Point to a Line If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. r ⊥ t & s ⊥t ⇒ r s The distance from a point to a line is the length of the perpendicular segment from the point to the line. The distance from A to the line t is AB rrrr ssss tttt AAAA BBBB tttt
- 4. Example Solve to find x and y in the diagram. 10x = 90 x = 9 8(9) + 4y = 90 4y = 18 y = 4.5 (8x+4y)° (10x)°
- 5. Distance From a Point to a Line 1. Name the shortest segment from R to FO. Since RG ⊥ FO, RG is the shortest segment 2. Write and solve an inequality for x. FR > RG x + 5 > 9 x > 4 F G O R x + 5x + 5x + 5x + 5 9999