Follow the hint given in the book, draw a diagram in the coordinate plane.
See solution.
Practice makes perfect
Let's follow the hint given in the book. We will plot the points on a coordinate plane and draw the triangles.
It looks that the two triangles are similar isosceles triangles.
We can use the Distance Formula to calculate the length of the sides to verify this observation.
Endpoints
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Simplify
A(0,0) and B(2,4)
sqrt(( 2- 0)^2+( 4- 0)^2)
A B=2sqrt(5)
B(2,4) and C(4,2)
sqrt(( 4- 2)^2+( 2- 4)^2)
B C=2sqrt(2)
C(4,2) and A(0,0)
sqrt(( 0- 4)^2+( 0- 2)^2)
C A=2sqrt(5)
R(0,3) and S(-1,5)
sqrt(( -1- 0)^2+( 5- 3)^2)
R S=sqrt(5)
S(-1,5) and T(-2,4)
sqrt(( -2- (-1))^2+( 4- 5)^2)
S T=sqrt(2)
T(-2,4) and R(0,3)
sqrt(( 0- (-2))^2+( 3- 4)^2)
T R=sqrt(5)
Now that we know the side lengths, we can write and simplify the ratio of corresponding side lengths. Let's start with the ratio of the shortest sides in the two triangles, which are BC and ST.
The ratio of the shorter sides of the triangles is 21. Note that the larger sides of â–ł ABC have the same length 2sqrt(5). Moreover, the larger sides of â–ł RST also have the same length sqrt(5). Therefore, to calculate the ratio of the larger sides of the triangles, it is enough with calculating just one of them. Let's find ABRS.
The ratio of the larger sides is also 21. We can see that the ratio of all three corresponding sides is the same.
AB/RS=BC/ST=CA/TR=2/1
By the Side-Side-Side Similarity Theorem, â–ł ABC and â–ł RST are similar. The scale factor is 2:1.
