Answer to Question #134995 in Trigonometry for pauwsa

Question

The area of a rhombus is 143m2. If the longer diagonal is 26m,
find the length of one side of the rhombus and its altitude.

Answer 1

The area of any convex quadrilateral can be determined as $S = \frac{1}{2}d_1 d_2 \sin\varphi$ , where $d_1$ and $d_2$ are the diagonals, $\varphi$ is the angle of diagonals' intersection (90 degrees for a rhombus).

Thus, the formula simplifies to $S = \frac{1}{2}d_1 d_2$. From here, $d_2 = \frac{2S}{d_1}=11$ m.

Then, if we take one quarter of a rhombus, we will have a right triangle with diagonals' halves as shorter sides of triangle and side of the rhombus as hypotenuse.

Hence, side of the rhombus a = $\sqrt{13^2+5.5^2}=14.11559\approx14.12$ m. <---------------------------

The last thing to calculate is the altitude h. From $S=a\cdot h$ we get $h=\frac{S}{a}=10.13064\approx10.13$ m. <-----------------------------