Perpendicular Lines

Definition

Perpendicular lines (also known as orthogonal lines) are a pleasure to work with, because they make things 'right'. There are several cases in which having a right angle greatly simplifies the problem. A simple application of right angles is the Pythagorean Theorem, which states that in a right triangle ABC ABC, where c c is the length of the hypotenuse and a a and b b are the lengths of the legs, then a2+b2=c2 a^2+b^2= c^2. This is a formula that everyone should be familiar with.

Another useful fact about perpendicular lines is that if we know that two triangles are right-angled, then the 'SSA similar triangles' hold. The reason for this, is that the ambiguous Sine Rule case is now ruled out.

In coordinate geometry, another useful property is that 2 lines are perpendicular if and only if the slope of one is the negative reciprocal of the other (ignoring case where slope = 0). Note that graphing would not help with determining perpendicular lines, unless an accurate diagram is drawn to scale.

The idea of orthogonality also extends to other areas of math, especially in Linear Algebra and Vector Spaces. For example, we say that 2 vectors are perpendicular if their dot product is 0, i.e. x,y=0 \langle x, y \rangle = 0, which generalizes to n n dimensions.

Technique

If AB AB is perpendicular to BC BC, and AB AB is perpendicular to BD BD, then points B,C B, C and D D lie on the same line.

If points C C and D D lie on the same side of AB AB, then since ABC=ABD \angle ABC = \angle ABD, hence B,C B, C and D D lie on the same line. This doesn't require the angle to be right. If points C C and D D lie on opposite sides of AB AB, then since ABC+ABD=90+90 \angle ABC + \angle ABD = 90^\circ + 90^\circ, hence CBD CBD is a straight line. _\square

 

PQR PQR is a right angled triangle with QPR=90 \angle QPR = 90^\circ. From P P, drop a perpendicular to QR QR intersecting at S S. Show that triangles PQR,SQP,SPR PQR, SQP, SPR are similar. Hence show that SQSR=SP2 SQ \cdot SR = SP^2.

By angle chasing, QPS=90PQS=90PQR=QRP \angle QPS = 90^\circ - \angle PQS = 90^\circ - \angle PQR = \angle QRP. Similarly, RPS=90SRP=90QRP=PQR \angle RPS = 90^\circ - \angle SRP = 90^\circ - \angle QRP = \angle PQR. Thus, by PPP, these three triangles are similar. By similarity, QSSP=SPSR \frac {QS}{SP} = \frac {SP}{SR}, which gives QSSR=SP2 QS \cdot SR = SP^2. _\square

 

Show that ABCD AB \perp CD if and only if AD2+BC2=AC2+BD2 AD^2 + BC^2 = AC^2 + BD^2. There is no requirement that line segments AB AB and CD CD intersect.

This problem is better presented using vector notation. Let the origin be O O (doesn't matter where). Let the points A,B,C,D A, B, C, D have position vectors a,b,c,d \vec{a}, \vec{b}, \vec{c}, \vec{d} . Then, AD2=ad,ad=a2+d22a,d AD^2 = \langle a-d, a-d \rangle = |a|^2 + |d|^2 - 2 \langle a,d \rangle . BC2=bc,bc=b2+c22b,c BC^2 = \langle b-c, b-c \rangle = |b|^2 + |c|^2 - 2 \langle b,c\rangle . AC2=ac,ac=a2+c22a,c AC^2 = \langle a-c, a-c \rangle = |a|^2 + |c|^2 - 2 \langle a,c \rangle . BD2=bd,bd=b2+d22b,d BD^2 = \langle b-d,b-d \rangle = |b|^2 + |d|^2 - 2 \langle b,d \rangle .

Hence, 2ba,dc=AD2+BC2AC2BD2 2 \langle b-a, d-c \rangle = AD^2 + BC^2 - AC^2 - BD^2 , which shows that ABCD AB \perp CD if and only if AD2+BC2=AC2+BD2 AD^2 + BC^2 = AC^2 + BD^2. _\square

Note: Students who are unfamiliar with vectors should prove this statement using the Cosine Rule (which is actually what we're using). Slight care has to be taken with regards to where AB AB intersects CD CD.

Note by Arron Kau
5 years, 2 months ago

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