Question

In triangle $XYZ$, $\angle X = 60^\circ$ and $\angle Y = 45^\circ$. The bisector of $\angle X$ intersects $\overline{YZ}$ at $W.$ If $XW = 24,$ then find the area of triangle $XYZ$.

[asy]
pair A,B,C,T,X,Y;

A = (0,0);
C = rotate(60)*(1,0);
B = (0.5+sqrt(3)/2,0);
T = intersectionpoint (C--B, A -- (rotate(30)*(5,0)));
draw(T--A--C--B--A);
label("$X$",A,SW);
label("$Y$",B,SE);
label("$Z$",C,N);
label("$W$",T,NE);
[/asy]

Answer 1

Answer:

We have,

ABCD as the cyclic quadrilateral in which the diagonal AC and BD.

intersect each other at point P.

also, given that,

AB=8cm,

CD=5cm

Now,

In ΔDCA and ΔAPB,

We have

∠DCP=∠ABP

∠CDP=∠PAB

Hence, 

ΔDPC∼ΔAPB (by A.A property)

According to the given question,

arΔAPBarΔDPC=(ABDC)2

⇒24arΔDPC=(85)2

⇒24arΔDPC=6425

⇒arΔDPV=6425×24

arΔDPC=9.375cm2