How to solve area of rhombus given one diagonal and an angle?

Question

Given PATS is a rhombus where diagonal PT = 6 cm and angle SPA=138. Determine the area of PATS .

Answer 1

Area of rhombus = 1/2 * product of lengths of diagonals. We know length of diagonal PT. We need to find length of diagonal AS.

The diagonals of a rhombus are perpendicular forming 4 right triangles. We will look at triangle at the bottom of the rhombus.

Because diagonals of a parallelogram bisect each other, half of PT (3) forms one leg of that right triangle and half of AS forms the other leg.

Because diagonals of a rhombus bisect each pair of opposite angles of a rhombus and < SPA = 138, the angle inside our right triangle is 69.

Let's label 1/2 of AS as x. tan 69 = x/3 --> x = 7.815, so AS = 15.63

So the area = 1/2 * 6 * 15.63 = 48.69 cm^2

Alternatively, since you have 4 congruent triangles you can compute area of triangle we are focusing on = 1/2 * 3 * 7.815 = 11.7225, then multiply by 4 to get 48.68 cm^2.

Answer 2

rhombus ke side equal hote hai , and digonal 90 degree per bisect kerte hai so ek digonal 6 hai to 3=3hoga and dushra digonal 4=4 ka hoga ,,when two digonal intersect at 90 degree then 4 eual trian formed so each triangle triplet formed 3,4,and 5 so 1st are of triangle 6 then area of 4 triangle is 6multiple4=24