Can you find area of the Green shaded Triangle? | (Polygon) | |

[00:00:00]Welcome to Primath. In this video, we have got this uh fivesided polygon A B C D E that consists of three triangles.

[00:00:10]This uh triangle A B E. This uh right triangle B C D and this uh green shaded tiny triangle B D E. As you can see in this given diagram such that the area of this uh triangle uh A B E has been given to us as 66 cm square. The area of this uh right triangle B C D has been given to us as 112 cm square as well. And moreover this side a b is 11 cm. Whereas this side a is 13 cm. And furthermore these angles this angle this angle and this angle are our 90° angles. And now our task is to calculate the area of this uh green shaded triangle B D A. Please don't forget to give a thumbs up and subscribe and please keep in mind that this figure may not be 100% true to the scale. Let's go ahead and get started. And here's our very first step. We know that we have been given the area of this uh triangle as 66 cm square and the area of this uh right triangle is 112 cm square as well.

[00:01:38]And we are interested in calculating the area of this uh green shaded triangle BD. So therefore our task is going to be calculate this side uh D A this side uh C D and likewise we are going to calculate uh this side uh B C as well and now we are going to think outside the box to make our job simple. So therefore we are going to draw some auxilary lines. As you can see in this uh next step I have created uh this right triangle A E F by extended uh these sides A B and uh this side uh D E.

[00:02:28]And this is our 90° angle. And as a result we are ended up with this uh rectangle uh F b C D. And now we are going to focus on this uh triangle A B E whose area has been given to us as 66 cm square. And now let's recall the area of a triangle formula.

[00:02:52]Area is always equal to a half time base time the height of the triangle. And we can see the base of this uh triangle AB E is this side AB which is 11 cm.

[00:03:06]Whereas the height of this uh triangle is E F and this side is unknown and the area has been given to us as 66. Let's go ahead and fill in the blanks in this uh formula. So our area of the triangle is 66 is going to be equal to a half * our base is 11 * our height is this side e f now I can write this one as 11 / 2 * e f is going to be equal to 66 6. And now I'm going to multiply both sides the reciprocal of 11 divid by 2 which is going to be 2 / 11. And I'm going to multiply on the right hand side with the 2 / 11 as well. And here we can see 2 / 11 and 11id by two they are gone. So therefore our EF segment length is going to be equal to 12 cm. So therefore we could see that this uh EF length turns out to be 12 cm.

[00:04:28]And here's our next step. Let's focus on this uh right triangle E F A. We know it's side lengths are 12 and 13. and this side uh a f is unknown. So therefore we are going to apply the pyagorean theorem on this triangle. And here's our pagan theorem. A square + b square = c square. And in our case our hypotenuse is 13. Wherever the two other legs are 12 and this side af. Let's go ahead and fill in the blanks in this pyagon formula. So we got uh a f² + 12 squar is going to be equal to 13².

[00:05:12]Let's simplify further more. A f² + 144 = 169. And now we are going to subtract 144 from both side. This is gone. So therefore a f² value turns out to be 25. Let me undo this square by taking square root on both sides. So therefore our side af length is going to be equal to positive 5 cm. So therefore our this uh side af length has got to be 5 cm and also let's make an observation. We could see that this uh five 12 and 13 our pyagorean triplets as well. And now let's make an observation. We can see that this whole side of this rectangle BF is going to be 11 + 5 is going to give us 16 cm. So therefore we conclude that this uh other side uh CD has got to be 16 cm as well.

[00:06:27]And now we are going to focus on this uh right triangle B CD whose area has been given to us as 112 cm square and it's uh one of the side lengths is 16 and this side uh B C is unknown. Let's recall the area of a triangle formula once again.

[00:06:48]Area of the triangle is always equal to a half time base time the height of the triangle. And in our case uh the base of this right triangle BCD is uh this side where is the height is this side uh BC and the area has been given to us as 112. Let's go ahead and fill in the blanks in this uh formula. So the area of this triangle has been given to us as 112 = to a half * our base is 16 and our height is bc length. And now we can see this uh 1 / 2 * 16 is going to give us 8. Let's divide both sides by 8. And we can see this 8 and 8 is gone. So therefore our B C side length turns out to be equal to 14 cm. So therefore our this uh BC side length turns out to be 14 cm.

[00:07:58]And now let's make an observation. We can see that this uh BC length is 14. So therefore this whole side uh DF is going to be 14 cm as well and we know that this uh segment uh EF length is 12. So therefore this tiny segment length has got to be 14 - 12 is going to give us 2 cm. So thus uh this uh segment uh de length turns out to be two. And here's our final step. Now we are going to calculate uh the area of this uh green shaded triangle B D. And now let's recall the area of triangle formula once again. Area is equal to a half time base time the height of the triangle. And now we can see that the base of this uh green shaded triangle is this uh tiny base ED which is 2 units. Whereas the height of this uh green shaded triangle is uh this length which is 16 cm. So therefore this uh green shaded triangle area is going to be a half time our base is 2 * the height is 16 and here we [clears throat] can see two and 1id by two they are gone. So this is going to be equal to 16 cm². the area of this green shaded triangle. So thus after all the calculations and manipulations the area of this uh green shaded triangle turns out to be 16 cm square. And that's our final answer. Thanks for watching and please don't forget to subscribe to my channel for more exciting videos. Bye.