Question

which of the following statements is true?
a. \\( \triangle j k l \cong \triangle m n o \\)
b. \\( \triangle j k l \cong \triangle p q r \\)
c. \\( \triangle s t u \cong \triangle m n o \\)
d. \\( \triangle s t u \cong \triangle j k l \\)

Explanation:

Step1: Calculate the side lengths of each triangle using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(\triangle JKL\):
- \(JK\): \(J(-8,-7)\), \(K(-3,-3)\)
\(JK=\sqrt{(-3 + 8)^2+(-3 + 7)^2}=\sqrt{25 + 16}=\sqrt{41}\)
- \(KL\): \(K(-3,-3)\), \(L(-2,-5)\)
\(KL=\sqrt{(-2 + 3)^2+(-5 + 3)^2}=\sqrt{1+4}=\sqrt{5}\)
- \(JL\): \(J(-8,-7)\), \(L(-2,-5)\)
\(JL=\sqrt{(-2 + 8)^2+(-5 + 7)^2}=\sqrt{36 + 4}=\sqrt{40} = 2\sqrt{10}\)

For \(\triangle MNO\):
- \(MN\): \(M(-2,7)\), \(N(-5,1)\)
\(MN=\sqrt{(-5 + 2)^2+(1 - 7)^2}=\sqrt{9+36}=\sqrt{45}=3\sqrt{5}\)
- \(NO\): \(N(-5,1)\), \(O(-5,4)\)
\(NO=\sqrt{(-5+5)^2+(4 - 1)^2}=3\)
- \(MO\): \(M(-2,7)\), \(O(-5,4)\)
\(MO=\sqrt{(-5 + 2)^2+(4 - 7)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\)

For \(\triangle PQR\):
- \(PQ\): \(P(6,0)\), \(Q(3,6)\)
\(PQ=\sqrt{(3 - 6)^2+(6 - 0)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}\)
- \(QR\): \(Q(3,6)\), \(R(3,2)\)
\(QR=\sqrt{(3 - 3)^2+(2 - 6)^2}=4\)
- \(PR\): \(P(6,0)\), \(R(3,2)\)
\(PR=\sqrt{(3 - 6)^2+(2 - 0)^2}=\sqrt{9+4}=\sqrt{13}\)

For \(\triangle STU\):
- \(ST\): \(S(-1,-6)\), \(T(1,0)\)
\(ST=\sqrt{(1 + 1)^2+(0 + 6)^2}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\)
- \(TU\): \(T(1,0)\), \(U(2,-4)\)
\(TU=\sqrt{(2 - 1)^2+(-4 - 0)^2}=\sqrt{1 + 16}=\sqrt{17}\)
- \(SU\): \(S(-1,-6)\), \(U(2,-4)\)
\(SU=\sqrt{(2 + 1)^2+(-4 + 6)^2}=\sqrt{9+4}=\sqrt{13}\)

For \(\triangle JKL\) and \(\triangle STU\):
- \(JK\) and \(ST\): \(JK=\sqrt{41}\), \(ST = 2\sqrt{10}\)
- \(KL\) and \(TU\): \(KL=\sqrt{5}\), \(TU=\sqrt{17}\)
- \(JL\) and \(SU\): \(JL = 2\sqrt{10}\), \(SU=\sqrt{13}\)

For \(\triangle JKL\) and \(\triangle MNO\):
- \(JK=\sqrt{41}\), \(MN = 3\sqrt{5}\)
- \(KL=\sqrt{5}\), \(NO = 3\)
- \(JL=2\sqrt{10}\), \(MO=3\sqrt{2}\)

For \(\triangle JKL\) and \(\triangle PQR\):
- \(JK=\sqrt{41}\), \(PQ = 3\sqrt{5}\)
- \(KL=\sqrt{5}\), \(QR = 4\)
- \(JL=2\sqrt{10}\), \(PR=\sqrt{13}\)

For \(\triangle STU\) and \(\triangle JKL\):
- \(ST = 2\sqrt{10}\), \(JL=2\sqrt{10}\)
- \(SU=\sqrt{13}\), \(KL=\sqrt{5}\)
- \(TU=\sqrt{17}\), \(JK=\sqrt{41}\)

Step2: Check for congruence using SSS (Side - Side - Side) criterion

For \(\triangle STU\) and \(\triangle JKL\):
- \(ST = JL=2\sqrt{10}\)
- \(SU\) and \(KL\) are not equal, \(TU\) and \(JK\) are not equal.

For \(\triangle JKL\) and \(\triangle MNO\):
- Sides are not equal.

For \(\triangle JKL\) and \(\triangle PQR\):
- Sides are not equal.

For \(\triangle STU\) and \(\triangle JKL\):
- \(ST = JL = 2\sqrt{10}\), \(SU\) and \(KL\) are not equal, \(TU\) and \(JK\) are not equal.

Let's use another approach. Count the number of units for vertical and horizontal segments (for right - angled parts if applicable).
For \(\triangle JKL\):
- \(JK\): horizontal change \(=|-3-(-8)| = 5\), vertical change \(=|-3-(-7)| = 4\) (length \(\sqrt{5^{2}+4^{2}}=\sqrt{41}\))
- \(KL\): horizontal change \(=|-2-(-3)| = 1\), vertical change \(=|-5-(-3)| = 2\) (length \(\sqrt{1^{2}+2^{2}}=\sqrt{5}\))
- \(JL\): horizontal change \(=|-2-(-8)| = 6\), vertical change \(=|-5-(-7)| = 2\) (length \(\sqrt{6^{2}+2^{2}}=\sqrt{40}=2\sqrt{10}\))

For \(\triangle STU\):
- \(ST\): horizontal change \(=|1-(-1)| = 2\), vertical change \(=|0-(-6)| = 6\) (length \(\sqrt{2^{2}+6^{2}}=\sqrt{40}=2\sqrt{10}\))
- \(TU\): horizontal change \(=|2 - 1| = 1\), vertical change \(=|-4-0| = 4\) (length \(\sqrt{1^{2}+4^{2}}=\sqrt{17}\))
- \(SU\): horizontal change \(=|2-(-1)| = 3\), vertical change \(=|-4-(-6)| = 2\) (length \(\sqrt{3^{2}+2^{2}}=\sqrt{13}\))

For \(\triangle MNO\):
- \(MN\): horizontal change \(=|-5-(-2)| = 3\), vertical change \(=|1 - 7| = 6\) (length \(\sqrt{3^{2}+6^{2}}=\sqrt{45}=3\sqrt{5}\))
- \(NO\): vertical segment of length \(|4 - 1| = 3\)
- \(MO\): horizontal change \(=|-5-(-2)| = 3\), vertical change \(=|4 - 7| = 3\) (length \(\sqrt{3^{2}+3^{2}}=\sqrt{18}=3\sqrt{2}\))

Answer:

D. \(\triangle STU\cong\triangle JKL\)