Question

note : figure not drawn to scale
in quadrilateral klmn shown kl = 3, kn = 27 and mn = 27. diagonals km and ln(not shown) intersect at point g(not shown), where gk = 1 and gm = 1. if the length of diagonal ln is √p + √w, where p and w are integers, what is the value of p+w?

Explanation:

Step1: Use the property of intersecting diagonals

In a quadrilateral, we can use the power - of - a - point theorem. Consider the intersection of diagonals $KM$ and $LN$ at $G$.

Step2: Apply the power - of - a - point for point $G$ with respect to $\triangle KLN$ and $\triangle LMN$

Let $GL = x$ and $GN=y$. By the power - of - a - point theorem: In $\triangle KLN$, $GK\cdot GM=GL\cdot GN$. Since $GK = GM = 1$, we have $GL\cdot GN = 1$. Also, using the distance formula in $\triangle KGL$ and $\triangle MGN$ with the given side - lengths. In $\triangle KGL$, by the Pythagorean theorem, $KL^{2}-GK^{2}=GL^{2}$, so $GL=\sqrt{KL^{2}-GK^{2}}=\sqrt{9 - 1}=\sqrt{8}$. In $\triangle MGN$, $MN^{2}-GM^{2}=GN^{2}$, so $GN=\sqrt{MN^{2}-GM^{2}}=\sqrt{27^{2}-1^{2}}=\sqrt{729 - 1}=\sqrt{728}$.

Step3: Calculate the length of $LN$

$LN=GL + GN=\sqrt{8}+\sqrt{728}=2\sqrt{2}+2\sqrt{182}$. So $p = 8$ and $w = 728$.

Step4: Find $p + w$

$p + w=8+728=736$.

Answer:

736