What do the given expressions tell about triangle △ PQS?
x=16 y=8
Practice makes perfect
Let's use colors to indicate corresponding vertices in the given congruence.
△ P Q S≅△ R Q S
Let's also put these colors on the figure.
Finding y
Notice that in triangle △ P Q S the expressions P Q=2x and P S=x indicate that P Q is twice as long as P S.
P Q=2 P S
Because of congruence, this relationship should also hold in triangle △ R Q S between the corresponding segments.
R Q=2 R S
Since the length of both of these segments are expressed in terms of y, this observation allows us to set up and solve an equation for y.
Notice that in the congruent triangles △ P Q S and △ R Q S segments P S and R S correspond to each other, so their measure is the same.
P S= R S
Since we now know that y=8, this will give us the value of x.
We found that if the two triangles on the figure are congruent, then x=16 and y=8.
Alternative Solution
Alternative way of thinking
After identifying corresponding vertices, we can identify corresponding sides.
Since the lengths of corresponding sides in congruent triangles are the same, this will allow us to write two equations in x and y.
Corresponding Sides
Equation
P Q and R Q
2x=3y+8
P S and R S
x=2y
We can find x and y by solving this system of equations.
