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?

Question

Noah Atkinson · Accepted Answer

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#

Wyatt Anderson · Answer

undefined We can determine if two triangles are similar by checking if their corresponding angles are congruent and their corresponding sides are proportional. In this case, we can compare the ratios of the lengths of the sides of DeltaMNP to the lengths of the sides of DeltaMQR. If the ratios are equal, then the triangles are similar. We can use the distance formula to calculate the lengths of the sides and then compare the ratios. If the ratios of the corresponding sides are equal, and the corresponding angles are congruent, then the triangles are similar.