Question

2 which statements are correct about the triangles below? select all that apply. □ de ≅ lm □ fd ≅ nl □ ef ≅ mn □ △def and △lmn are not congruent

Explanation:

To solve this, we use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the lengths of the sides of triangles \(DEF\) and \(LMN\).

Step 1: Find coordinates of vertices

• Triangle \(DEF\): \(D(10, 0)\), \(E(2, 1)\), \(F(-3, 2)\)
• Triangle \(LMN\): \(L(3, 1)\), \(M(5, -4)\), \(N(-5, -5)\)

Step 2: Calculate length of \(DE\) and \(LM\)

• For \(DE\): \(x_1 = 10, y_1 = 0\); \(x_2 = 2, y_2 = 1\)

\[
DE = \sqrt{(2 - 10)^2 + (1 - 0)^2} = \sqrt{(-8)^2 + 1^2} = \sqrt{64 + 1} = \sqrt{65}
\]

• For \(LM\): \(x_1 = 3, y_1 = 1\); \(x_2 = 5, y_2 = -4\)

\[
LM = \sqrt{(5 - 3)^2 + (-4 - 1)^2} = \sqrt{2^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}
\]
Wait, that can't be right. Wait, maybe I misread the coordinates. Let's re - check the coordinates from the graph:

Looking at the graph again, let's correctly identify the coordinates:

For triangle \(DEF\):

• \(D\): Let's assume the grid is such that each square is 1 unit. From the graph, \(D\) seems to be at \((5,0)\) (maybe my initial reading was wrong), \(E\) at \((2,1)\), \(F\) at \((- 3,2)\)

For triangle \(LMN\):

• \(L\) at \((3,1)\), \(M\) at \((5, - 4)\), \(N\) at \((-5,-5)\)

Wait, maybe a better approach is to look at the answer options:

Option 1: \(DE\cong LM\)

Let's recalculate with correct coordinates. Let's assume:

\(D=(5,0)\), \(E=(2,1)\), \(F=(-3,2)\)

\(L=(3,1)\), \(M=(5,-4)\), \(N=(-5,-5)\)

Calculate \(DE\):

\(x_1 = 5,y_1 = 0\); \(x_2=2,y_2 = 1\)

\(DE=\sqrt{(2 - 5)^2+(1 - 0)^2}=\sqrt{(-3)^2 + 1^2}=\sqrt{9 + 1}=\sqrt{10}\)

Calculate \(LM\):

\(x_1 = 3,y_1 = 1\); \(x_2 = 5,y_2=-4\)

\(LM=\sqrt{(5 - 3)^2+(-4 - 1)^2}=\sqrt{2^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29}\). No, that's not equal.

Wait, maybe the coordinates of \(D\) is \((10,0)\), \(E=(2,1)\), \(F=(-3,2)\) and \(L=(3,1)\), \(M=(5,-4)\), \(N=(-5,-5)\) is wrong. Let's check the other options.

Option 2: \(FD\cong NL\)

Calculate \(FD\): \(F(-3,2)\), \(D(10,0)\)

\(FD=\sqrt{(10+3)^2+(0 - 2)^2}=\sqrt{13^2+( - 2)^2}=\sqrt{169 + 4}=\sqrt{173}\)

Calculate \(NL\): \(N(-5,-5)\), \(L(3,1)\)

\(NL=\sqrt{(3 + 5)^2+(1 + 5)^2}=\sqrt{8^2+6^2}=\sqrt{64 + 36}=\sqrt{100} = 10\). Not equal.

Option 3: \(EF\cong MN\)

Calculate \(EF\): \(E(2,1)\), \(F(-3,2)\)

\(EF=\sqrt{(-3 - 2)^2+(2 - 1)^2}=\sqrt{(-5)^2+1^2}=\sqrt{25 + 1}=\sqrt{26}\)

Calculate \(MN\): \(M(5,-4)\), \(N(-5,-5)\)

\(MN=\sqrt{(-5 - 5)^2+(-5 + 4)^2}=\sqrt{(-10)^2+( - 1)^2}=\sqrt{100+1}=\sqrt{101}\). Not equal.

Option 4: \(\triangle DEF\) and \(\triangle LMN\) are not congruent.

To check congruence, we can check the side lengths. Since we saw that the side lengths we calculated are not equal (e.g., \(DE=\sqrt{(2 - 10)^2+(1 - 0)^2}=\sqrt{64 + 1}=\sqrt{65}\), \(LM=\sqrt{(5 - 3)^2+(-4 - 1)^2}=\sqrt{4 + 25}=\sqrt{29}\), \(EF=\sqrt{(-3 - 2)^2+(2 - 1)^2}=\sqrt{25 + 1}=\sqrt{26}\), \(MN=\sqrt{(-5 - 5)^2+(-5 + 4)^2}=\sqrt{100 + 1}=\sqrt{101}\), \(FD=\sqrt{(10 + 3)^2+(0 - 2)^2}=\sqrt{169+4}=\sqrt{173}\), \(NL=\sqrt{(3 + 5)^2+(1 + 5)^2}=\sqrt{64 + 36}=10\)), the triangles do not have all corresponding sides equal. So \(\triangle DEF\) and \(\triangle LMN\) are not congruent.

Answer:

\(\triangle DEF\) and \(\triangle LMN\) are not congruent (the option with the statement "\(\triangle DEF\) and \(\triangle LMN\) are not congruent")