Perpendicular Lines Definition Two intersecting lines will have four angles formed at the intersection points. If all the four angles are equal, then the two lines are said to be perpendicular to each other. We already know by linear postulate theorem that the two vertically opposite angles are equal. Hence if these two lines are perpendicular, then all four angles are 90 degrees. Examples of perpendicular lines: In the graph paper, The X-axis and Y-axis are perpendicular. In an ellipse two axes, minor axis, and major axis are perpendicular. For a line segment, any shortest line from a point outside the circle is perpendicular. Tangent and normal to any curve are perpendicular lines. Slopes of two perpendicular lines: In coordinate Geometry, when two lines are perpendicular, the product of the slopes of the lines is -1. This property has a lot of applications in finding the equation of perpendicular lines, length of perpendicular segment from a point to a given line, etc. Know More About Congruent Shapes Worksheets
For any curve in a graph with equation y = f(x), the slope of the tangent is defined as the rate of change of y with respect to x at that point. The normal to this curve at this point is perpendicular to the tangent line. Example: In a circle, with centre at the origin and radius 3, the equation will be of the form (x)²+(y)² = 3². Take any point say (0,3). To find the tangent, we have to find dy/dx. Differentiating, 2x+2y =0 Hence, the slope of the normal is perpendicular to x axis or parallel to y axis. Example for Perpendicular Lines from a Point to a LineBack to Top Let AB be a line with coordinates (1,2) and (3,4). Measure the length of perpendicular line from (-1,1) to this line segment. We know that the perpendicular line from (-1,1) has a slope of -1/slope of AB. Equation of AB is (x-1)/(3-1) = (y-2)/(4-2) Or x-1 = y-2 Or y = x+1 Slope of AB passing through (1,2) and (3,4) is 4 - 2/3 -1 =1. Slope of perpendicular line to AB is -1. Learn More Counting Money Worksheet
Since the perpendicular line passes through (-1,1) equation of the perpendicular is y-1 = -1(x+1) or y =-x -1 +1 or y = -x. To get the foot of the perpendicular line on AB, we solve the two equations by substitution method. y = x+1 = -x This on simplification gives 2x = -1 or x = -1/2. Since y = -x , we have y = +1/2, So, foot of the altitude from the point (-1,1) is (-1/2,1/2). The length of the perpendicular segment is between (-1,1) and (-1/2,1/2) is √[ (-1/2+1)²+(1/2-1)²] = √(1/4+1/4) = √(1/2) = 1/1.414 = 0.707 approximately.
Pythagorean Theorem Examples The Pythagorean theorem is related to the study of sides of a right angled triangle. It is also called as pythagoras theorem. The pythagorean theorem states that, In a right triangle, (length of the hypotenuse)2 = {(1st side)2 + (2nd side)2}. In a right angled triangle, there are three sides: hypotenuse, perpendicular and base. The base and the perpendicular make an angle of 90 degree with eachother. So, according to pythagorean theorem: (Hypotenuse)2 = (Perpendicular)2 + (Base)2 Euclid Proof of Pythagorean Theorem According to Euclid, if the triangle had a right angle (90 degree), the area of the square formed with hypotenuse as the side will be equal to the sum of the area of the squares formed with the other two sides as the side of the squares.
From the above figure 3, the sum of the area covered by the two small squares is equal to the area of the third square. Here, a2 is the area of the square ABDE, b2 is the area of the square BCFG and c2 is the area of the square ACHI. Therefore, a2 + b2 = c2 Hence Proved. Pythagorean Theorem Example Problems Below are example problems on Pythagorean theorem Example Problem 1: In a right triangle, the hypotenuse is 5 cm and the perpendicular is 4 cm. Find the length of the base of the triangle? Solution: By using Pythagoras theorem, h2 =p2 + b2 52 = 42 + b2 25 = 16 + b2
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9 = b2 b=3 Base is 3 cm Example Problem 2: In a right triangle, the base is 6 cm and the perpendicular is 8 cm. Find the length of the hypotenuse of the triangle? Solution: By using Pythagoras Theorem, h2 =p2 + b2 h2 = 62 + 82 h2 = 36 + 64 h = 10 Hypotenuse is 10 cm
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