Question

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? note: figure not drawn to scale.

Explanation:

Step1: Use Pythagorean - theorem in right - triangles

In right - triangle $KGL$, $KL = 3$, $KG = 1$, by $a^{2}+b^{2}=c^{2}$, $GL=\sqrt{KL^{2}-KG^{2}}=\sqrt{9 - 1}=\sqrt{8}$. In right - triangle $MGN$, $MN = 3$, $MG = 1$, $GN=\sqrt{MN^{2}-MG^{2}}=\sqrt{9 - 1}=\sqrt{8}$.

Step2: Calculate length of $LN$

$LN=GL + GN=\sqrt{8}+\sqrt{8}=2\sqrt{8}=\sqrt{32}+\sqrt{0}$, so $p = 32$, $w = 0$, and $p - w=32$.

Answer:

32