The length of the shorter altitude and the shorter leg of parallelogram are 9 cm and squared sqrt(82 cm. The length of the longer diagonal is 15 cm. What is the area of this parallelogram?
The shorter altitude or simply the height of the parallelogram and the shorter side are sides of a right triangle. √82 is the hypotenuse. So, we can find the other side using the Pythagorean theorem
(√82)^2 = 9^2 + x^2
x = 1 cm (the length of the other side and part of base length)
We are also given the length of the longer diagonal. When you draw, you will see that it forms a right triangle if we extend a straight line towards the base. The diagonal is the hypotenuse. The distance from the right angle to the parallelogram's long side will also be 1 cm because of congruent triangles formed according to ASA postulate (this part is hard to explain with words. If you want to see my drawing, you can message me). And the shorter side will be the height, 9 cm. We can find the other side using the Pythagorean theorem
15^2 = 9^2 + y^2
y = 12 cm.
However, this includes the extra 1 cm length. So, the base length is 12-1 = 11 cm
Now, we know the base and the height so the area is
A = base*height
9*11 = 99 cm^2