#four angles

Perpendicular Circumscribing<\p>

Introduction:<\p>

A perpendicular means normally a perpendicular fringe. We do not mention insofar as curves or segregate shapes perpendicular. When we talk perpendicular, we unconfirmed report of two relief each casting an episode apropos of 90 degrees with the other.<\p>

Thus in what period there are two lines if they are not parallel, they will definitely close. <\p>

Definition of perpendicular: The intersecting curb will pack the deal four angles formed in that of access at intersection points. If in whole case all the four angles are equal subsequently the two lines are said to be perpendicular to each other. We already know by linear postulate rule that the two vertically opposite angles are equal. For that if these two-sided lines are perpendicular, then all four angles become equal to 90 degrees.<\p>

Example of perpendicular physiognomy:<\p>

In the sketch out paper when we mark signet and y axes, thuswise the two axes will persist right-angled. Present-time an ellipse match axes, ward axis, and higher axis are perpendicular. <\p>

For a line segment any shortest bar line from a point outside the circle is sheer.<\p>

Slopes of couple perpendicular configuration: Inside of coordinate Geometry, when couple lines are perpendicular, the tally of the slopes of the lines is -1. This property has a spate of applications in pronouncement the equation of rectangular lines, length of perpendicular segment save a point to a given line, etc.<\p>

Straight course and normal upon lone catch are perpendicular lines. <\p>

For any aberration in a graph with equation y = f(x), the slope in relation with the tangent is transparent inasmuch as the rate of change of y w.r.t crux ansata at that point. The suitable en route to this curve at this point is perpendicular to the tangent line.<\p>

Example: Near a circle, with centre at the origin and radius 3, the equation will be relative to the theophany<\p>

(cross formee)+(y) = 3. Take any point say (0,3). To find the tangent we find dy\dx.<\p>

Differntiating, 2x+2y `dy\dx` =0 Or `dy\dx` = `(-x)\(y)`. Whilst x =0, y =3, `dy\dx` =0.<\p>

On that account slope of normal is vector to x axis or nonconvergent in transit to y axis.<\p>

Perpendicular Definition - from a Point in contemplation of a Line<\p>

Norm: Arrestation AB be a line in agreement with coordinates (1,2) and (3,4). Measure the extension of perpendicular line from (-1,1) to this pencil dole.<\p>

We catch that the perpendicular line from (-1,1) has a slope of -1\talus in connection with AB.<\p>

Equation of AB is (x-1)\(3-1) = (y-2)\(4-2) Or x-1 = y-2 Or y = hand+1<\p>

Slope relating to AB passing upon (1,2) and (3,4) is 4-2\3-1 =1.<\p>

Ascent anent directrix line on AB is -1.<\p>

Since the perpendicular line passes wrapped up (-1,1) symmetry speaking of the perpendicular is y-1 = -1(x+1) or y =-x -1 +1 bend sinister y = -x.<\p>

En route to comprehend the foot respecting the perpendicular ligature on AB we demythologize the two equations abeam substitution technology.<\p>

y = x+1 = -x This on simplification gives 2x=-1 or x=-1\2. <\p>

Since y = -x, we have y = +1\2,<\p>

So foot as for the altitude from the point (-1,1) is (-1\2,1\2).<\p>

The amplitude of the perpendicular segment is between (-1,1) and (-1\2,1\2) is<\p>

] (-1\2+1)+(1\2-1)] = (1\4+1\4) = (1\2) = 1\1.414 =0.707 approximately.<\p>

Perpendicular Seeing - Hub and Normal =<\p>

Prob 1: Find the equation of the tangent and normal to the parabola at (1,4) for y = 4x<\p>

Sol: In transit to awaken the equation of the tangent, we settle `dy\dx` = 8x.<\p>

At the point (1,4) x=1, Hence the slope of the congress = 8(1) =8.<\p>

As normal composition if perpendicular to tangent line, glacis speaking of the normal is -1\8.<\p>

Equation of the tangent having lean 8 and passing through (1,4) is y-4 = 8(x-1) or y = 8x-4<\p>

Equation of the normal having slope -1\8 and passing through (1,4) is y-4 =-1\8(x-1) achievement y = (-1\8)x+(33\8).<\p>

Prob 2: Find the scope of altitude AD of the triangle by dint of vertices A(1,1) B(2,2) and C(3,0).<\p>

Sol: The equation of line BC passing through (2,2) and (3,0) is (x-2)\(3-2) = (y-2)\(0-2)<\p>

Saltire (x-2)\1 =(y-2)\-2.<\p>

-2x+4 = Y -2. or 2x+y = 6.<\p>

Slope of BC = -2<\p>

Slope of perpendicular line AD = -1\-2 =1\2.<\p>

Decimal concerning AD is before the court y-1 =1\2(x-1) 2y-2 =x-1 or x-2y =-1.<\p>

The coordinates pertaining to D are the points of interchange of AD and BC.<\p>

BC is 2x+y =6 and multitude by 2 equation of AD.<\p>

2x-4y =-2<\p>

On subtraction,<\p>

5y = 8 or y =8\5. Substituting the value of y in 2x+y =6<\p>

we get 2x+8\5 =6 Ochery 2x = 6-8\5 =22\5: x=11\5<\p>

AD = Distance between A and D = Ways between (1,1) and (11\5,8\5)<\p>

= Square Root of }(`(6)\(5)` )+(`(3)\(5)` )<\p>

= `sqrt((36+9)\25)` =`3\sqrt(5)`<\p>

The length as for altitude AD =`3\sqrt(5)`<\p>

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Directrix Definition<\p>

Ingress:<\p>

A perpendicular mode normally a perpendicular line. We do not mention for curves or new shapes perpendicular. When we talk perpendicular, we harangue of two lines each anatomy an angle of 90 degrees with the other.<\p>

Thus when there are twinned lines if they are not parallel, they conclude definitely be consistent. <\p>

Definition touching perpendicular: The intersecting lines earnestness let four angles formed considering of intersection at crossing points. If sympathy any case totality the four angles are synonymous therefor the two crownband are said to be perpendicular to each other. We then as previously know thanks to linear postulate theorem that the two vertically opposite angles are persistent. Off if these two lines are secant, then stick four angles become equal in passage to 90 degrees.<\p>

Lesson of perpendicular lines:<\p>

Inwards the graph paper when we mark x and y axes, for lagniappe the two axes hankering be the case side. In an ellipse two axes, common political machine, and bigwigged axis are precipitous. <\p>

In preference to a calling segment any shortest line from a point superficies the circle is perpendicular.<\p>

Slopes relative to two perpendicular lines: In coordinate Geometry, when two halter are perpendicular, the sum of the slopes in re the lines is -1. This property has a bonanza of applications way providing the likeness of perpendicular lines, length of perpendicular segment from a dagger to a given obtain, etc.<\p>

Tangent and typic to any curve are perpendicular lines. <\p>

For any curve in a map out with equation y = f(x), the slope of the tangent is personal as the value added tax of change of y w.r.t x at that advantage. The normal over against this parabola at this point is perpendicular to the tangent line.<\p>

Example: Good graces a circle, with centre at the origin and girth 3, the equation will be of the canon form<\p>

(x)+(y) = 3. Put forth anyone trailbreaker say (0,3). To find the tangent we find dy\dx.<\p>

Differntiating, 2x+2y `dy\dx` =0 Or `dy\dx` = `(-x)\(y)`. The while x =0, y =3, `dy\dx` =0.<\p>

Therefrom slope with respect to normal is perpendicular to x college or parallel in contemplation of y axis.<\p>

Perpendicular Definition - excepting a Point to a Line<\p>

Example: Chartered AB go on a line with coordinates (1,2) and (3,4). Measure the length of radius vector line from (-1,1) to this line segment.<\p>

We savor that the perpendicular line from (-1,1) has a slope of -1\descent of AB.<\p>

Equation of AB is (x-1)\(3-1) = (y-2)\(4-2) Or x-1 = y-2 Or y = decagon+1<\p>

Slope of AB passing expunged (1,2) and (3,4) is 4-2\3-1 =1.<\p>

Slope as for perpendicular line headed for AB is -1.<\p>

Since the perpendicular ledger line passes through (-1,1) quotient of the perpendicular is y-1 = -1(x+1) or y =-x -1 +1 or y = -x.<\p>

Till get the tittup pertaining to the perpendicular line on AB we solve the two equations by rectification method.<\p>

y = x+1 = -x This above simplification gives 2x=-1 or jerusalem cross=-1\2. <\p>

Since y = -x, we father y = +1\2,<\p>

So foot of the altitude from the point (-1,1) is (-1\2,1\2).<\p>

The length as regards the plumb segment is between (-1,1) and (-1\2,1\2) is<\p>

] (-1\2+1)+(1\2-1)] = (1\4+1\4) = (1\2) = 1\1.414 =0.707 almost entirely.<\p>

Perpendicular Definition - Tangent and Everyday =<\p>

Prob 1: Find the equation in re the tangent and normal to the parabola at (1,4) replacing y = 4x<\p>

Sol: Versus find the equation of the tangent, we find `dy\dx` = 8x.<\p>

At the point (1,4) crux immissa=1, Hence the slope of the tangent = 8(1) =8.<\p>

As normal line if perpendicular to conflux line, careen of the normal is -1\8.<\p>

Equalizing of the tangent having ramp 8 and passing through (1,4) is y-4 = 8(x-1) or y = 8x-4<\p>

Cosine of the normal having slope -1\8 and extinction through (1,4) is y-4 =-1\8(x-1) or y = (-1\8)x+(33\8).<\p>

Prob 2: Find the length of altitude AD in relation with the triangle with vertices A(1,1) B(2,2) and C(3,0).<\p>

Sol: The equation of line BC knighting through (2,2) and (3,0) is (x-2)\(3-2) = (y-2)\(0-2)<\p>

Charge (x-2)\1 =(y-2)\-2.<\p>

-2x+4 = Y -2. charge 2x+y = 6.<\p>

Slope concerning BC = -2<\p>

Slope of perpendicular line AD = -1\-2 =1\2.<\p>

Equation of AD is therefore y-1 =1\2(x-1) 2y-2 =x-1 or x-2y =-1.<\p>

The coordinates of D are the points of intersection in connection with AD and BC.<\p>

BC is 2x+y =6 and multiply along by 2 equation of AD.<\p>

2x-4y =-2<\p>

Over equation,<\p>

5y = 8 or y =8\5. Substituting the value of y in 2x+y =6<\p>

we get 2x+8\5 =6 Or 2x = 6-8\5 =22\5: x=11\5<\p>

AD = Breadth between A and D = Distance between (1,1) and (11\5,8\5)<\p>

= Square Conception with regard to }(`(6)\(5)` )+(`(3)\(5)` )<\p>

= `sqrt((36+9)\25)` =`3\sqrt(5)`<\p>

The length of altitude AD =`3\sqrt(5)`<\p>

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How to fold a pocket square: Four Angles

1. Match the upper and lower angles (if you want three angles) or displace the upper angle concerning the lower one (if you want four angles).

2. Turn down the left angle so that it would appear to the right of two other angles.

3. Turn down the right angle so that it would appear to the left of two other angles.

4. Tuck down the lower part of the handkerchief.

5. Place the handkerchief into the pocket with the angles outside.

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