Given points #M(0, 10)#, #N(5, 0)# and #P(15, 15)# in #DeltaMNP# and points #M(0, 10)#, #Q(10, -10)#, and #R(30, 20)# in #DeltaMQR#, how do we find that the two triangles are similar?

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Answer 1 · 2017-01-07T14:47:13

Find all the sides and check whether they are proportional.

Given two points #A(x_1,y_1)# and #B(x_2,y_2)#, the distance between the two #AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

As we have points #M(0, 10)#, #N(5, 0)# and #P(15, 15)# in #DeltaMNP#

and points #M(0, 10)#, #Q(10, -10)#, and #R(30, 20)# in #DeltaMQR#

and we have in #DeltaMNP#

#MN=sqrt((5-0)^2+(0-10)^2)=sqrt(25+100)=sqrt125=5sqrt5#

#NP=sqrt((15-5)^2+(15-0)^2)=sqrt(100+225)=sqrt325=5sqrt13#

#MP=sqrt((15-0)^2+(15-10)^2)=sqrt(225+25)=sqrt250=5sqrt10#

and in #DeltaMQR#

#MQ=sqrt((10-0)^2+(-10-10)^2)=sqrt(100+400)=sqrt500=10sqrt5#

#QR=sqrt((30-10)^2+(20-(-10))^2)=sqrt(400+900)=sqrt1300=10sqrt13#

#MR=sqrt((30-0)^2+(20-10)^2)=sqrt(900+100)=sqrt1000=10sqrt10#

Therefore #(MN)/(MQ)=(NP)/(QR)=(MP)/(MR)=1/2#

and as sides of two triangles are proportional, we have

#DeltaMNP~~DeltaMQR#