Why Do Most Students Fail To Figure This Out?! Is It So Tricky!??

[00:00:00]In this figure, we are given a square which has side length two and inside the square we have this shaded region for whose area we required to find. We're going to start by labeling the points of this square. Let this be A, B, C, and this D.

[00:00:24]The next step is we're going to draw a diagonal from point A to point C.

[00:00:33]So just like that.

[00:00:41]So now we're going to consider three figures formed out of this. We're considering three. So the first one is this one here separated by this line.

[00:00:57]this one.

[00:00:59]This is A and this is C.

[00:01:04]Okay.

[00:01:05]And also we're going to look at this here where this is A, this is B and this is C.

[00:01:27]Also, we're going to look at the triangle formed.

[00:01:36]Okay, this is A, this is B and this is C.

[00:01:44]Now from here or from this we can see that this one here the area of this just to find the area of this one here this part of the shaded region it will be equals to the area of this here minus the area of the triangle.

[00:02:08]So this part of the shaded region is equals to this one here is a quarter circle minus this triangle.

[00:02:18]So let's start by finding the area of the quarter circle.

[00:02:28][snorts] So this is the quarter circle. This is a b and this is c.

[00:02:38]And the radius is B C which is 2.

[00:02:48]We know that area of a circle is p<unk> r² but since this is a quarter circle shall multiply by a4. Okay. Therefore area is equals to a4 *<unk> radius is 2. So 2^ 2. So this is equ= to 1 out of 4 *<unk> * 2 * 2. This cancels with this to give two. This cancels with this.

[00:03:22]So we are left with just * 1 * 1 which is just pi as the area of the quarter circle.

[00:03:32]Then for the area of the triangle, this one here.

[00:03:44][snorts] So for the triangle A B C since B C is two and also BA is also side side of the square which is also two. So area of this one is a half * base * height. Therefore area will be equals to a half time the base is 2 and the height is two. So this cancels with this and area is two square units for this triangle.

[00:04:26]Therefore, the area of the half shaded region we've said we said it area is equals to area of quarter circle minus area of triangle.

[00:04:52]Okay. So the area we found the area of the quarter circle to be pi and the area of the triangle we found it to be 2.

[00:05:00]Therefore area is equals to<unk> - 2 for the for only this one here for half of the shaded region. But since a c bisects this square into two equal parts that means that half of this shaded region is equals to the other half of the shaded region.

[00:05:23]Okay. Therefore, area [clears throat] of shaded region is equals to 2 * area of the half of the shaded region which is<unk> - 2.

[00:05:41]So area is equals to 2 * this 2 *<unk> is 2<unk> - 2 * 2 is 4 which is approximately equals to 28 square units.

[00:06:00]So [snorts] this is the area of the shaded region. In this figure we are required to find the value of angle x.

[00:06:09]So, we have this angle here is X. This angle is 40°. This angle is 20°. And they're telling us that this line is equals to this line here from here to here. So, with that information, let's find the value of angle X. Let's start by labeling the points on our figure.

[00:06:33]Let this be a b this C and let this be D.

[00:06:43]Now next let us extend this line DB B towards the left up to a point say E such that when we connect that point E to A, we form an angle of 20° at that point E.

[00:07:00]So let me do that.

[00:07:03]So when I extend this line say up to there and I'm calling this point E and then connect A to E.

[00:07:19]I should be able to make 20° here.

[00:07:26]I know the diagram is not to scale but I'm sure you get the point.

[00:07:32]Now that we can say that this angle is equals to this angle. So that implies that this side here A E is equals to this side A C.

[00:07:49]And so we can conclude and say triangle A E C is an isocles an isocles triangle.

[00:08:07]Okay.

[00:08:10]Now we know that from the exterior angle theorem the exterior angle of the exterior angle of a triangle.

[00:08:21]So if you have something like this exterior angle of a triangle say this one here is y which is the exterior angle. It is equal to the sum of the two opposite interior angles.

[00:08:42]So let's say this is alpha and this is beta. Okay. So from exterior angle theorem y is equals to alpha + beta.

[00:08:59]So following this in our diagram considering this angle 40 as the exterior angle that implies that this 40 will be equals to 20 plus this angle here this one.

[00:09:16]Okay. So that implies that angle a bd is equals to angle a e plus angle e a b.

[00:09:31]Okay. So we we have this angle is 40.

[00:09:37]Then e a e b is 20.

[00:09:43]Then this is the angle we don't have.

[00:09:47]Therefore, angle E A B will be equals to 40° minus 20° which gives us [clears throat] 20°.

[00:10:00]Therefore, this angle here is 20°.

[00:10:06]Now since in this triangle B A E this angle is 20° and also this angle is 20° that implies that this triangle is also an isocles triangle implying that this side AB is equals to side AB.

[00:10:26]Okay.

[00:10:28]Although the diagram is known to scale and with that according to side angle side theorem that implies that triangle E A B this one here is congruent to triangle C A D this one due to side angle side theorem. So as you can see this side is equals to this side. This angle is equals to this angle and this side is equals to this side.

[00:11:02]Now if this is true then since in this triangle here the angle that is uh that is corres that corresponds to the 20 the side that corresponds to angle this angle of 20° is side A.

[00:11:20]Okay. So that implies that uh the side that corresponds to the to the one that is equals to this in this triangle which is DC. So remember DC is equals to A B and the side that the angle that corresponds with A is 20.

[00:11:42]That implies that also in this triangle here the sides that the side that corresponds to DC is also 20° and that side is X. So that G means X is equals to 20°. Hope you've understood