Question

using the provided figure below, find the length of ln.

Explanation:

Step1: Identify the triangle properties

Since $\angle N = 90^{\circ}$ and $\angle MLN = 90^{\circ}$ (given the right - angle at $L$) and $\angle M = 45^{\circ}$ (because in $\triangle MNK$, $\angle K = 45^{\circ}$ as $\triangle MNK$ is a right - isosceles triangle, and the line from $L$ creates similar triangles). The triangle $\triangle MLN$ is a $45 - 45-90$ triangle. In a $45 - 45 - 90$ triangle, the sides are in the ratio $1:1:\sqrt{2}$. Also, note that $\triangle KLN$ is similar to $\triangle MNK$ (by AA similarity, as $\angle K$ is common and $\angle MLN=\angle N = 90^{\circ}$).
Let $LN=x$. Then $KN=\sqrt{2}x$ (from the $45 - 45 - 90$ triangle ratio in $\triangle MNK$). And $KL = 12$. Since $KN=KL + LN$, and $KN=\sqrt{2}x$, we have $\sqrt{2}x=x + 12$.

Step2: Solve the equation for $x$

Rearrange the equation $\sqrt{2}x=x + 12$:

$$
LATEXBLOCK0
$$

Rationalize the denominator: multiply numerator and denominator by $\sqrt{2}+1$

$$
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$$

Answer:

$29$ (approximate value, if we consider the $45 - 45 - 90$ triangle properties and solve step - by - step as above. If we assume another approach: Since $\triangle MLN$ is a $45 - 45 - 90$ triangle and $KL = 12$, and using the fact that in a right - isosceles triangle, if we consider the length relationships more intuitively (by the property that the segment from the right - angle to the hypotenuse in a right - isosceles triangle, and after proper geometric analysis and approximation) the value of $LN$ is approximately $29$)