Question

units 5 and 6 benchmark
select all the descriptions of the figure that must be true.
figure of a geometric diagram with right angles at k and m, segments labeled 3x - 4 and 5x + 12, and options: x = 8, kn = 24, mn = 52, kl ≅ ml, ln ≅ lm

Explanation:

Step1: Identify congruent triangles

Since \( \angle K = \angle M = 90^\circ \), \( \angle KLN=\angle MLN \) (given by the angle mark), and \( LN \) is common, triangles \( KLN \) and \( MLN \) are congruent by AAS. Thus, \( KN = MN \) and \( KL = ML \).

Step2: Solve for \( x \)

Set \( 7x - 4 = 5x + 12 \) (since \( KN = MN \)).
\( 7x - 5x = 12 + 4 \)
\( 2x = 16 \)
\( x = 8 \).

Step3: Check \( KN \) and \( MN \)

For \( KN \): \( 7(8) - 4 = 56 - 4 = 52 \).
For \( MN \): \( 5(8) + 12 = 40 + 12 = 52 \). So \( KN = MN = 52 \).

Step4: Check \( KL \cong ML \)

From congruent triangles (AAS), \( KL \cong ML \) is true.

Step5: Check \( LN \cong LM \)

\( LN \) is a leg, \( LM \) is a hypotenuse? No, \( LM \) is a leg? Wait, no—\( LN \) and \( LM \): \( LM \) is a side of triangle \( MLN \), but \( LN \) is common? Wait, no, \( LN \) is the same in both, but \( LM \) and \( LN \): from the diagram, \( LM \) is a leg, \( LN \) is a hypotenuse? Wait, no, actually, in congruent triangles \( KLN \) and \( MLN \), \( KL = ML \), \( KN = MN \), \( LN = LN \). But \( LN \) and \( LM \): \( LM \) is equal to \( KL \), not \( LN \). So \( LN \cong LM \) is false.

Answer:

• \( x = 8 \) (True)
• \( KN = 24 \) (False, \( KN = 52 \))
• \( MN = 52 \) (True)
• \( KL \cong ML \) (True)
• \( LN \cong LM \) (False)

So the true ones are \( x = 8 \), \( MN = 52 \), \( KL \cong ML \), and \( KN = 52 \) (wait, \( KN = 7x - 4 = 52 \), so \( KN = 52 \), not 24. So correct true statements: \( x = 8 \), \( MN = 52 \), \( KL \cong ML \), and \( KN = 52 \) (but the option is \( KN = 24 \) which is false). So the correct options are \( x = 8 \), \( MN = 52 \), \( KL \cong ML \), and \( KN = 52 \) (but the option for \( KN \) is 24, which is wrong). Wait, the options:

Options:

• \( x = 8 \) (True)
• \( KN = 24 \) (False)
• \( MN = 52 \) (True)
• \( KL \cong ML \) (True)
• \( LN \cong LM \) (False)

So the correct ones are \( x = 8 \), \( MN = 52 \), \( KL \cong ML \), and \( KN = 52 \) (but the option is \( KN = 24 \), so that's false). So the true descriptions are \( x = 8 \), \( MN = 52 \), \( KL \cong ML \).