Question

triangle jkl is defined by the coordinates j(5,3), k(8,3), and l(8,4).
triangle jkl was rotated 90° counterclockwise about the origin and translated 7 units left and 2 units up to create triangle tuv with
coordinates t(-4,-3), u(-4,-6), and v(-3,-6).
determine if these two triangles are congruent. if they are not congruent, state the reason why.
oa. the triangles are congruent.
ob. the triangles are not congruent because neither their corresponding sides nor their corresponding angles are congruent.
oc. the triangles are not congruent because their corresponding sides are not congruent.
od. the triangles are not congruent because their corresponding angles are not congruent.

Explanation:

Step1: Calculate side lengths of triangle \(JKL\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(JK\) with \(J(5,3)\) and \(K(8,3)\): \(d_{JK}=\sqrt{(8 - 5)^2+(3 - 3)^2}=\sqrt{3^2+0^2}=3\).
For \(KL\) with \(K(8,3)\) and \(L(8,4)\): \(d_{KL}=\sqrt{(8 - 8)^2+(4 - 3)^2}=\sqrt{0^2+1^2}=1\).
For \(JL\) with \(J(5,3)\) and \(L(8,4)\): \(d_{JL}=\sqrt{(8 - 5)^2+(4 - 3)^2}=\sqrt{3^2+1^2}=\sqrt{9 + 1}=\sqrt{10}\).

Step2: Calculate side lengths of triangle \(TUV\)

For \(TU\) with \(T(-4,-3)\) and \(U(-4,-6)\): \(d_{TU}=\sqrt{(-4+4)^2+(-6 + 3)^2}=\sqrt{0^2+(-3)^2}=3\).
For \(UV\) with \(U(-4,-6)\) and \(V(-3,-6)\): \(d_{UV}=\sqrt{(-3 + 4)^2+(-6+6)^2}=\sqrt{1^2+0^2}=1\).
For \(TV\) with \(T(-4,-3)\) and \(V(-3,-6)\): \(d_{TV}=\sqrt{(-3 + 4)^2+(-6 + 3)^2}=\sqrt{1^2+(-3)^2}=\sqrt{1+9}=\sqrt{10}\).

Step3: Check congruence

Since \(JK = TU = 3\), \(KL=UV = 1\), \(JL = TV=\sqrt{10}\), by SSS (Side - Side - Side) congruence criterion, the two triangles are congruent. So options B, C, D are incorrect.

Answer:

A. The triangles are congruent.