Question

The parallel sides of a trapezium are 25 cm and 13 cm; its nonparallel sides are equal, each being 10 cm. Find the area of the trapezium.

Answer 1

Measure of the parallel sides= 25 cm and 13 cm

Measure of non-parallel sides= 10  cm each

Height= unknown hence a variable is taken as 'x'

(The illustration of the trapezium is attached along, please check it out)

Area of trapezium= $\frac{1}{2}$× sum of parallel sides × height

The trapezium has two right-angled triangles. Hence, we'll use the Pythagoras theorem here.

Formula= $Height^{2} + Base^{2} = Hypotenuse^{2}$

So...

The height here is unknown so let it be 'x' (I mentioned that before). Base here should be divided and found (long process) and the hypotenuse would be 10 cm.

The base measure of the right-angled triangle (excluding the other parts of the line; don't take the whole base measure of the trapezium)

As u can view in the image attached, the opposite side, i.e., the side which measures 13 cm in the trapezium, shows that the side measure of the quadrilateral which is formed not including the right-angled triangles that were made by drawing the height; that is, the opposite side would also be 30. (check out the second image if my explanation ain't clear...

So... To find the base measure

Subtract 25 with 13 and then you'll get 12. Since there r two right-angled triangles and we're just dealing with one of them, we'll simply divide 12 by 2 since there are two right-angled triangles (I repeated that again). Anyways, so we get.... Voila! 6 cm!

Therefore, the base would be 6 cm.

So... coming back to what we were doing that is Pythagorean stuff-

We've got the base, hypotenuse but not the height. So let's put them all in place.

Height= x

Hypotenuse= 10 cm

Base= 6 cm

Therefore,

$x^{2} + 6^{2} = 10^{2}$

$x^{2} + 36= 100$

$x^{2} = 100 - 36\\x^{2} = 64\\x= \sqrt{64} \\x= 8 cm$

Hence, height equals 8 cm. Therefore, we got the height of the trapezium (Squee!).

Now, the final stage that is, the area. (dun dun daaaaa)

Area of trapezium= $\frac{1}{2}$ x (sum of parallel sides) x height

                             = $\frac{1}{2}$ x (13 + 25) x 8

                             = 152 $cm^{2}$ (The step is there in the 4th image attached)

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There! U got the ans! Happy learning! Hope it helps you mate! I'll be glad if it did! ;)