Question

determine whether ( \triangle abc cong \triangle klm ).
( a(-3,3), b(-1,3), c(-3,1), k(1,4), l(3,4), m(1,6) )
find the side lengths of each triangle.
a) ( ab = 2, kl = 2, bc = 2sqrt{2}, lm = 2sqrt{2}, ac = 2, km = 2 )
b) ( ab = 2, kl = 2, bc = sqrt{2}, lm = sqrt{2}, ac = 2, km = 2 )
c) ( ab = 2, kl = 2sqrt{2}, bc = 2sqrt{2}, lm = 2sqrt{2}, ac = 2, km = 2 )
d) ( ab = 2, kl = 2, bc = 2sqrt{2}, lm = 2sqrt{2}, ac = 2, km = 2sqrt{2} )

Explanation:

Step1: Calculate \(AB\) and \(KL\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(AB\) with \(A(-3,3)\) and \(B(-1,3)\):
\(AB=\sqrt{(-1+3)^2+(3 - 3)^2}=\sqrt{2^2+0^2}=2\)
For \(KL\) with \(K(1,4)\) and \(L(3,4)\):
\(KL=\sqrt{(3 - 1)^2+(4 - 4)^2}=\sqrt{2^2+0^2}=2\)

Step2: Calculate \(BC\) and \(LM\)

For \(BC\) with \(B(-1,3)\) and \(C(-3,1)\):
\(BC=\sqrt{(-3 + 1)^2+(1 - 3)^2}=\sqrt{(-2)^2+(-2)^2}=\sqrt{4 + 4}=2\sqrt{2}\)
For \(LM\) with \(L(3,4)\) and \(M(1,6)\):
\(LM=\sqrt{(1 - 3)^2+(6 - 4)^2}=\sqrt{(-2)^2+2^2}=\sqrt{4 + 4}=2\sqrt{2}\)

Step3: Calculate \(AC\) and \(KM\)

For \(AC\) with \(A(-3,3)\) and \(C(-3,1)\):
\(AC=\sqrt{(-3+3)^2+(1 - 3)^2}=\sqrt{0^2+(-2)^2}=2\)
For \(KM\) with \(K(1,4)\) and \(M(1,6)\):
\(KM=\sqrt{(1 - 1)^2+(6 - 4)^2}=\sqrt{0^2+2^2}=2\)

Answer:

A. \(AB = 2\), \(KL = 2\), \(BC = 2\sqrt{2}\), \(LM = 2\sqrt{2}\), \(AC = 2\), \(KM = 2\)