How To Solve A Hard Geometry Problem That Stumps Many | Area of Rectangle? #geometrypuzzle #geometry

Added: 2026-05-10

[00:00:00]Welcome amazing one and let's solve this beautiful geometry puzzle together. We are given a right triangle and inscribed is a rectangle such that the side of the rectangle lies on this part and on this part of the right triangle and also touches the right triangle at this point.

[00:00:21]From here to here measure seven units and this side of the right triangle from here to here measures 12 units. The question is asking us to find the area of this rectangle, the shaded. Is this something you can do? Don't worry, I will take you through it step by step.

[00:00:39]Now, what we need to do is to find the area of the rectangle what you need to do is remember area of a rectangle is given as so we have it that area of rectangle is given as the length multiplied by the width. So, if we label from here to here X, if we label this length X units and then label this length Y units then it means that the area will be to multiply these two sides. So, we have X * Y and that gives us the area. But the question is do we know X and do we know Y? Until we find these two, then we can find the area of the rectangle. Let me show you how you can do it. Is it possible for us to find the area of the bigger shape? The bigger shape is the right triangle. If we can find the area of the bigger shape then we now find the area if you watch closely you'll see remember this is a rectangle. It means that the angle formed here is 90°.

[00:01:48]And this is also 90. In that case, it means that the angle formed here is also 90° because this is on a straight line.

[00:01:56]And if this is true, you see that this part of the shape has given you a right triangle because of this right angle.

[00:02:03]So, it means that if you also look at this part, you also observe that here is 90°. The remaining angle is also going to be 90 because it's still on a straight line. And in that case, we have also formed a right triangle here. So, it means that if you want to find the area of this rectangle, all we need to do is so we can have it that area of rectangle is equal to the area of the bigger shape, which is area of the bigger triangle, minus the area of this smaller triangle.

[00:02:40]So, we can call it shape one, then we call this one shape two.

[00:02:46]Then we have the area of the smaller triangle, which is triangle one, minus the area of this one, which is area of triangle two. Until we do this, then it gives us the area of the rectangle. Now, the question is how do we find the area of this bigger triangle, this bigger one? How do we find it? You know that the area of a triangle is always given as the base times the height divided by two. So, we're going to use this rule to work it out. Let's do it together. And if you are returning viewer, thank you for always choosing to learn tutorials.

[00:03:22]If you are still new, why not subscribe for the more you get? Make sure the notification bell is on because you get updates from us every day. Don't forget to also like and share. Thank you, and let's keep doing this to have So, we have it that for the triangle for the bigger triangle, we have area of this triangle is going to give us Using this, we are going to have This is the height, and this is the base. So, we multiply these two lengths.

[00:03:48]From here to here is going to give us Y plus 12.

[00:03:54]And from here down to this end, we give us X plus 7. So, it means that the area of the bigger triangle will give us area will give us X plus 7 times Y plus 12.

[00:04:10]And when you multiply this, you divide by two, obeying what you have stated here. So, what should we do? We are going to open up the bracket. So, we have area of triangle will give us Use this to multiply. So, multiply this and also do same. X times X is X. X times Y is XY. X times 12 gives us 12 X. Then go with this.

[00:04:34]7 times Y is 7 Y. Then 7 times 12 gives us 84.

[00:04:41]So, what should we do? Remember, it's divided by two.

[00:04:44]And this is the same as area of that triangle is the same as XY over two plus 12 X over two plus 7 Y over two then 84 over two. This is for the area of the bigger triangle. Now, let's keep it aside and get the area of triangle one and then area of triangle two. Let's do it together. To find area of triangle one, we're going to have So, area of triangle one is going to give us This is what we have. This is the height. From here to here is the base. And you see that for a rectangle, the opposite sides are equal.

[00:05:31]So, from here to here is Y. This will also be Y. So, we have that area of triangle one will give us the these two lengths multiplied which is 7 times Y divided by two using the formula. And that gives us area of triangle one is 7y over two.

[00:05:51]Now that we have found this area, we can now find the area of triangle two. For triangle two, from here to here is the same with this and it's the same as x. From here to here is 12. So, area of triangle two will give us this time this which is 12 * x. So, we have 12 multiplied by x divided by two. And that gives us area of triangle two is 12x over two. So, this is what we have for area of triangle two. Now that we have found area of triangle, the bigger triangle, triangle one and two, let's bring them together to find the area of the rectangle. Remember, this is what we have for area of triangle, triangle one and triangle two. So, we're going to have area of the rectangle will now be the area of triangle minus this as we stated.

[00:06:47]So, work this out, we are going to have area of the rectangle will be the area of triangle which is xy plus over two plus 12x over two plus 7y over two plus 84 over two then subtract area of triangle one which is 7y over two.

[00:07:15]Subtract area of triangle two is 12x over two.

[00:07:19]So, let's subtract this. This will subtract this, it's gone.

[00:07:23]Then 7y over two subtract this, it's gone.

[00:07:26]And we are left with area of triangle, so we can work it out here. Area of rectangle is going to give us So, this gives us xy over two then plus So, we can adjust this.

[00:07:45]So, we have area of rectangle is equal to xy over two plus 84 over two.

[00:07:57]So, what should we do? So, we have area of rectangle is equal to The LCM is the same, so it means x + y over 84 divided by two. But, remember that the rectangle we have is given as This is our rectangle. We are looking for its area. And the area is going to be the product of these two sides as we said. So, we said area of the rectangle is xy. So, in that case, it means we are going to replace it here. So, let's work it out here.

[00:08:30]So, to now have We are going to now have area of rectangle we replace with xy.

[00:08:36]So, we have xy will give us x y + 84 divided by two. So, what should we do?

[00:08:45]We just need to multiply both sides by two so that we can clear this.

[00:08:50]So, this takes off.

[00:08:52]Multiply, we have 2xy is equal to xy + 84.

[00:08:58]To remove this, you subtract xy from both sides.

[00:09:02]And when you do that, we can now conclude here. So, now have Subtract this, we have xy is equal to This will give us 84. So, it means that the area of our rectangle, as we have said, is going to give us 84 squared units. So, this becomes our answer.

[00:09:26]So, this is our answer for finding the area of this rectangle.

[00:09:31]And I hope you learned something. Don't forget to tell us how much. Like, share.

[00:09:36]I also be curious to see your own method. Thank you for always choosing to learn tutorials. I look forward to seeing you in my next video. Have a lovely day. Bye.