The triangles AJKL and AQRS are not congruent based on the Side-Side-Side (SSS) criterion. The ratios of the corresponding side lengths are not equal, indicating that the triangles do not have the same shape and size.
To determine whether triangles AJKL and AQRS are congruent, we can use the Side-Side-Side (SSS) congruence criterion. According to the SSS criterion, if the lengths of the corresponding sides of two triangles are equal, then the triangles are congruent.
Given the coordinates of the vertices:
J(1, -7), K(2, -5), L(1, 0), Q(-2, 8), R(−3, 5), S(2, 1)
To check for congruence, we need to compare the lengths of the corresponding sides:
1. Side AJ and side AQ:
AJ = sqrt((2-1)^2 + (-5-(-7))^2) = sqrt(1^2 + 2^2) = sqrt(5)
AQ = sqrt((2-(-2))^2 + (1-8)^2) = sqrt(4^2 + 7^2) = sqrt(65)
2. Side JK and side QR:
JK = sqrt((2-2)^2 + (-5-0)^2) = sqrt(0^2 + 5^2) = 5
QR = sqrt((-3-(-2))^2 + (5-1)^2) = sqrt((-1)^2 + 4^2) = sqrt(17)
3. Side KL and side RS:
KL = sqrt((1-2)^2 + (0-(-5))^2) = sqrt((-1)^2 + 5^2) = sqrt(26)
RS = sqrt((2-(-3))^2 + (1-5)^2) = sqrt(5^2 + (-4)^2) = sqrt(41)
Now, let's compare the lengths of the corresponding sides:
AJ/AQ = sqrt(5)/sqrt(65)
JK/QR = 5/sqrt(17)
KL/RS = sqrt(26)/sqrt(41)
Since the ratios of the side lengths are not equal, AJKL and AQRS are not congruent triangles.