Question

in a triangle PQR Q=90 and QM is an altitude. if PQ=8and QR= 6,find area of triangle QMR

Answer 1

Let PQ=xPQ=x, QR=yQR=y and PR=zPR=z.

Given: x+y+z=60x+y+z=60 (i);
Equate the areas: 12∗xy=12∗QS∗z12∗xy=12∗QS∗z (area of PQR can be calculated by 1/2*leg*leg and 1/2* perpendicular to hypotenuse*hypotenuse) --> xy=12zxy=12z (ii);
Aslo x2+y2=z2x2+y2=z2 (iii);

So, we have:
(i) x+y+z=60x+y+z=60;
(ii) xy=12zxy=12z;
(iii) x2+y2=z2x2+y2=z2.

From (iii) (x+y)2−2xy=z2(x+y)2−2xy=z2 --> as from (i) x+y=60−zx+y=60−z and from (ii) xy=12zxy=12z then (60−z)2−2∗12z=z260−z)2−2∗12z=z2 --> 3600−120z+z2−24z=z23600−120z+z2−24z=z2 --> 3600=144z3600=144z --> z=25z=25;

From (i) x+y=35x+y=35 and from (ii) xy=300xy=300 --> solving for xx and yy --> x=20x=20 and y=15y=15 (as given that x>yx>y).

Next, perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. So, PQS and SQR are similar. In two similar triangles, the ratio of their areas is the square of the ratio of their sides: AREAarea=S2s2AREAarea=S2s2.

So, x2y2=AREAareax2y2=AREAarea --> AREAarea=400225=169AREAarea=400225=169