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: Apply Pythagorean theorem in right - angled triangles formed by diagonals.

In right - angled triangles $\triangle KGL$ and $\triangle GMN$, if we consider the right - angled triangles formed at the intersection of diagonals. Let the diagonals $KM$ and $LN$ intersect at $G$. We know that in right - angled triangle, if we assume the segments of the diagonals form right - angled triangles.
Let $KG = x$, $GL=y$, $GM = z$, $GN = w$. By Pythagorean theorem, $LN^{2}=(KG + GM)^{2}+(GL + GN)^{2}$. Since $KG = 1$ and $GM = 1$, and using the fact that we can consider the right - angled structure of the quadrilateral's diagonals.
$LN=\sqrt{(1 + 1)^{2}+(0+0)^{2}}=\sqrt{4}=2$. Since $LN=\sqrt{p}+\sqrt{w}$ and $LN = 2$, we can assume $\sqrt{p}=\sqrt{1}$ and $\sqrt{w}=\sqrt{1}$, so $p = 1$ and $w = 1$.

Step2: Calculate $p + w$.

$p+w=1 + 1=2$

Answer:

2