Students in this video, let's discuss the questions that were part of the geometry quiz now before
you watch this video, make sure that you have attempted all the questions in the quiz.
All right.
Let's get started with the first question.
The first question says three identical circles are tangent to each other.
This shaded region is 36 square root of three, minus 18 by find the area of the three circles taken
together.
All right.
But we have to find the area of these three circles.
What's the answer to this question?
The correct answer is option E that is under an eight by now.
How do we approach this?
It's given that these three circles are identical.
So let them be of radius are therefore this distances are and these distances also are similarly over
here you have R and R and here also you have R and R.
So triangle ABC is an equilateral triangle with each side equal to two R right now.
The area of an equal triangle is given by Route three by four in two sides square.
The site over here is two, so that's Route three by four.
In the two are square, which is equal to room three are square.
All right.
Now, this shaded region over here would be the area of this triangle minus the area of this segment,
minus the area of this segment, minus the area of the segment.
All right.
Now, because there's an equal triangle, these angles will be 60 degrees right now.
Then we can easily find the area of this segment.
Right.
That would be 60 by 316 to Pyar Square, because if the angle is 360 degrees.
The area of the segment is Byaah Square.
That is the whole circle right now.
Therefore, if the angle is 60 degrees, the area would be Pyar Square, divided by 360 into 60.
That's what we have over here.
Now, we have three such segments that we multiply with three.
Therefore, the shaded area is route three are square minus three to sixty by 360 by our square.
This simplifies to Route three are square minus Pyar Square by two.
And it's given that this is equal to 36 through three minus 18 by now.
Over here on solving, you can see that our square is equal to 36.
Right.
And over here you have our square by two.
That gives you 18.
Therefore our square is equal to 36 to our is equal to six.
Therefore, the area of one circle will be Pyar Square, which is right into 36.
The area of all three circles taken together is three in the square, which is equal to one hundred
and eight by which is your option eight.
Let's read the next question in the figure below, C is equal to is equal to any one could draw a circle
with Base B as the center, OK?
And having A and B on the circle find angle BDA.
This is the angle that we need to find, all right, but one could draw a circle with B as the center
and having E and the name on the circle.
All right.
Now, the answer to this question is option B. That's 72 degrees.
Now let's see how to do this.
It's given that BK's equal to SC.
Therefore, these two angles will be equal because BKE is isosceles triangle.
Similarly, it's also given that X equal to 80 A.D., an isosceles triangle.
Therefore, these two angles will be equal.
That's Angle X and angle ADC.
All right.
Now you can see that X is equal to Y plus Y, right?
Because this angle here is an exterior angle for triangle ABC, therefore X, Y plus Y, because the
exterior angle is equal to the sum of the interior opposite angles.
All right.
So we have to wait now over here.
You can see that angle, b, b, e, d, this angle over here will be equal to and will be the B, the.
Why is that so.
Because B and we have to be equal because they are really a right vs the center of the circle and and
the light on the circle.
So B is radius similarly.
B is it radius.
Alright now you can see that Y this way over here, plus 180, minus two X.
That's this angle over here.
Right.
So this plus this, that's this angle over here has to be equal to this angle which is X because we
as equal to beauty and therefore be it is an isosceles triangle but these two are equal.
Now that simplifies to Y plus one is equal to three X.
Now we know that Y plus one, it can be written at six Y because X is equal to two way.
Right.
So three X is equal to six Y and on simplifying you will get one.
It is equal to five or Y is equal to 180 by five which is equal to thirty six.
And hence you can find X, which is what we need to find because X is equal to white.
That's 236, which is equal to seventy two degrees.
And that is your option B.
Let's read the next question in the figure below ABCDE CEF is a regular hexagon, the perimeter of Triangle
BATF is equal to six Route three.
All right.
Find the perimeter of the hexagon, so we need to find the perimeter of the hexagon.
But what's the answer to this question?
The correct answer to this question is option B. That's 12.
Now, remember, this is a regular hexagon.
So we have seen in a previous video that you can split it into six equilateral triangles like this right
now in equilateral triangle A, B, or BP is the height, right.
And height of any greater triangle is giving us Route three by two in two sides right now.
B, if you can see, is two times BP, because BP and B, will be equal B, B, the height of an equal
Ranglin.
If B is also the height of the triangle, or if the B.F. is equal to two times BP, that's equal to
Route three.
All right, so that's the side of the equilateral triangle over here, so similarly, Beauty and D.F.
is also Route three and the perimeter of this triangle, BATF is equal to three times Route three,
and that is given a six Route three right in the question.
Therefore, we get that is equal to two.
Right.
And if is equal to do is the side of the hexagon.
Therefore, the perimeter of the hexagon will be six in the two.
That is equal to twelve.
And that is option B.
Let's read the next question.
Find the length of a diagonal through the center of a regular hexagon offsite, ten centimeter.
All right.
What's the answer to the question?
The answer to this question is two times, then that is 20 centimeter.
All right.
Again, we have our hexagon over here and we have split it into six equilateral triangles.
Now, over here, you can see that B is A diagonal, which is passing through the center of the hexagon.
Right.
And you can see that B or is equal to or is equal to side of the hexagon.
Right.
And the side is then therefore B is equal to two times the side of the hexagon.
That's 20 centimeter, which is option A.
Let's read the next question, BQ artists are calling your points the line on the x axis is nearer to
the origin.
Q is second nearest to the origin of his third nearest to the origin and is farthest from the origin
p.
Q is three centimeter pieces, tends to be R seven centimeter and Q R is what we need to find.
Try to draw a diagram when you get these type of questions.
But what's the answer to this question.
The correct answer is option B, that's four centimetre.
Right.
You can just imagine that the origin is somewhere over here.
Now in the question we can find that the order of the boyens is in this manner because it's given PS
and to the origin, to his second nearest to the origin, etc..
Now it's given that P Q is three centimetres P R is seven centimeters.
Right.
And we need to find QR.
So QR will be P, R minus P, Q that seven minus three which is equal to four centimetre.
That is option B.
Let's read the next question, the volume of a cylinder is three eight eight five six centimetre cube
express this in millimeter cube.
All right.
So what's the correct answer to this question?
The correct answer is option eight, because one centimetre cube is one centimeter into one centimeter
into one centimeter.
We know that one centimeter is equal to 10 millimeter.
Right.
So in zero one centimeter, 10 millimeter over here also we have 10 millimeter.
And over here also we have 10 millimeter.
When you multiply these, you get a thousand millimetre cube.
So one centimetre cube is equal to a thousand millimetre cube.
Therefore, three eight eight five six centimetre cube is equal to create five three eight eight five
six in two thousand millimeter cube.
That gives you the answer as option A.
Let's read the next question in the figure ABCDE and FDX asks, where's the ratio of 80 of ABCDE to
the area in between the two squares is 36, is 220.
That is the area of ABCDE to the area that's in between.
That is this part of the year.
This ratio is 36 is 220, the site of about 10 centimeters.
This side over here find itself.
All right.
What's the answer to this question?
The answer to this question is option B, that is 20 by three centimeter.
Now let's take the ratio of area of ABCDE and area of.
ABC minus 80 of FGF, that's what is given us, that is by 20, 36 is the area of ABC and ABC minus
EVGA.
It will give you this intermediate area, and that's 36 by 20.
Now, if the area of ABC is 36 X, then the area of FGF will be thirty six X minus the intermediate
area.
That's thirty six X minus 20 X.
That's equal to 16 X.
All right, now let's take the ratio of the areas of the two squares that that's 36 X by 16 X.
That will be equal to the ratio of the squares of the corresponding sides, which is A square by B square
right to ABI.
B is equal to six by four because this X and this gets cancelled.
So Abi B is six by four and we know that is equal to ten because that's given and F is what.
We need to find out, so a baby six by four, the article written by F. NCF is equal to 10 in the two
by three, which is equal to twenty by three centimeters, which is option B.
Let's proceed to the next question.
In the hexagon ABCDE, Abe is equal to four centimeter, all right, and BKS five centimeters is six
centimeter, DEA is five centimeter, F is seven centimeter.
Which of the following cannot be the length of if not, the figure is not drawn to scale.
All right.
What's the answer to this question?
The answer to this question is option E, that's twenty seven centimetre.
Right.
For this we join EMC and D and E and E.
Now over here we have a triangle ABC and hence we know that some of two sides of a triangle has to be
greater than the third side.
Therefore AC has to be less than four plus five, which is equal to nine.
All right.
Now, over here, this side is less than nine.
That's what we just now found out.
Therefore, it has to be less than nine plus six, which is 15 to 80 is less than nine plus.
That is it is less than 15.
So let me write that over here now.
Therefore, E will be less than twenty.
Right.
That's fifteen plus five for E has to be less than twenty and in a similar manner if has to be less
than twenty seven.
Right.
So if has to be less than twenty seven.
Therefore if cannot be twenty seven centimetre.
That's what we have to find out.
Right.
Therefore the answer to this question is option eight.
Let's read the next question in a triangle, ABC, the site BKS centimeter the length of the altitude
through a two sided B.C. is equal to one centimeter.
All right.
So whenever you have these questions, draw a rough sketch.
You have triangle ABC.
All right.
Now, it's given that the length of the altitude through a two sided BKK is equal to Y centimeters.
So this is Y centimeter.
And it's given that the side BKS X centimeter, the area of triangle ABC is equal to two square centimeters.
All right.
So that means half into X and Y is equal to the middle square.
Which of the following indicates the possible values of Y if X is greater than light?
All right.
So what is the answer to this question?
The answer to this question is option C, that is why it has to be less than two.
Right.
Now, let's discuss why that is.
So now over here, X and Y have to be positive as they are parts of a triangle.
Right.
But we don't need to take any negative values into consideration.
All right.
Now, it is half in the base in the height area.
That's two is equal to half into X, Y, so X Y is equal to four.
So this is what we get from the given information.
But from this, I can say that X is equal to four by right now.
It's given that X is greater than Y before by Y has to be greater than Y or four is greater than Y square
or Y is less than two.
So over here we don't take negative values because they are parts of a triangle.
The Y has to be less than two and the correct answer is option C.
Let's read the next question and Circle asks or as center, and we are two points on the circle such
that the length of archabbey is split into 10 by three.
It's given that the radius of the circle extends centimeter.
Find the length of the cord a bit.
All right.
But what's the answer to this question?
The answer to this question is option C, that is 10 centimeters.
Right here we have our circle, this the center, and there are two points on the circle.
Now, the length of the arc is given us by ten, by three.
Right now, the circumference of the circle is to Pyar.
That's to buy into ten because it's given that the radius is ten.
Now if you take a circle to pi are right, the angle is 360 degree.
Now if you have a length of arc as pi in ten by three, which is the length of the arc Abe, then the
degree angle will be 360 by power into pi into ten by three to 360 by two.
By ten into pi entertained by three and that simplifies to 60 degrees.
Therefore we get that this angle over here.
Is equal to 60 degrees, right, therefore, B.
Is also equal to the radius, right?
If this is 60 degrees and or is equal to O.B., so these two angles have to be equal, that means this
is also 60 extra degree and this is also 60 degrees and triangle or maybe is an equilateral triangle
right now.
This is the radius, this the radius.
And in an extended triangle, all sides are equal.
Therefore, Abby also has to be equal to the radius.
And it's given that the radius is equal to 10 centimeter.
Therefore, the correct answer to this question is option C.
Let's read the next question.
Find the shaded area in centimeter square.
Now it's given that ABCDE is a rectangle, but we need to find this shaded region already.
What's the answer to this question?
The correct answer is option C, that is 100 centimetres square.
Now the shaded area is equal to the area of the rectangle minus the area of the triangle.
Right Triangle PDC area of rectangle ABCDE extending to 20.
That's been 20.
Over here, an area of Triangle Park is half baked into height.
That's half 20 to ten because this height over here also will be 10 centimeters.
Right.
That's this height itself.
So that's half 2013 and that's 10 to 20 minus 20 by two, which is equal to 10 to 20 by two.
That's hundred centimetre square.
That's option C or here.
All right, let's read the next question, Clara wants a house equidistant from the church, school
and hospital.
She plots these three in a map using the coordinate system as shown in the figure.
Where should she buy her house?
Because she wants to buy her house in a manner that renovated houses, for example, for houses over
here.
Then this distance, this distance and this distance has to be equal.
Now, what's the answer to this question?
The answer to this question is option E.
You just need to find the sock'em center right, because with the Sagami center as the center, you
can draw a circle that passes through these three points.
And in that case, this is equal to the radius.
This is equal to the radius.
And this part also is equal to the radius.
Right now.
Over here, you have a right angle triangle, right?
Because these two axis, you know, are at 90 degree angle with each other.
Now, in a right angle triangle, it's very easy to find the circumstance that it will be the midpoint
of the hypotenuse right to over here.
The midpoint is seven plus zero by two.
That's seven by two and 11 plus zero by two.
That's one by two.
Therefore, the Sock'Em Center is seven by two column, eleven by two.
And that is the point where she should buy her house, which is option A.
Let's read the next question, find angle, a plea, a plea, that's this angle over here, plus a QB,
that's this angle over here, plus B or C plus C SD.
So we need to find this angle plus this angle plus this angle plus this angle.
All right.
So what's the answer to this question?
The correct answer to this question is option C, that's 360 degrees right now.
Over here is an interesting problem.
We know that nine percent plus 11 plus 12 is equal to 360 degree because ABCDs is a quadrilateral right
now.
Over here, you can also see that one plus nine plus eight is equal to 180 degree because this is a
straight angle, right?
Similarly, two plus ten plus three is also a 180 degree or plus eleven plus five is 180 degree.
Seven plus 12 plus six is 180 degree.
So if you add this up, you have 180 in two for now.
Over here you have nine, 10, 11 and 12.
And we know that when you add them, you get 360.
But we take that out.
And the remaining part, one plus eight plus two plus three plus four plus five plus six plus seven
will be one 18 to four minus 360, which is nothing but 360 because we are it to forestry, 16 to 360
to minus 360 gives you 360.
All right.
Now over here, these are triangles right there for one plus two.
Plus this angle over here is 180 degree.
Eight plus seven plus angle PS 180 degree six plus five plus angle S is 180 degree and three plus four
plus angle or is equal to 180 degrees.
So we write that over here.
If you add these up, that's 118 to four because there are four triangles.
Now over here, you can see that these parts are there eight, seven, one, two, three, four, six,
five.
These are the eight parts over here.
And we know that they add up to 360 degree.
So we take that out and we are left with people as Q Plus our places is equal to 316 to minus 360,
which is equal to 360 degree.
And that is your option C.
Let's read the next question mark bought a bed that is in the shape of a square.
All right.
He bought a bed sheet that is in the shape of a circle.
The bed sheet is such that it hangs out excellently on the four sides of the bed.
The side of the bed is to be does find the area of the bed sheet.
All right.
What's the answer to this question?
The answer is option B, that is to paint.
This is a case where a circle is circumscribing square.
Right.
You can see that over here, this much part will be hanging out of the bed sheet similarly on this side.
This part will be hanging out, right.
It's equal on all sides.
That's why the circle is circumscribing this square.
All right.
Now, over here, you can see that AC is a diagonal of the square as well as diameter of the circle.
Right over here.
This is two.
This is also true.
Therefore, diagonal of the square will be root of two squared plus two square, which is equal to two.
That's equal to the diameter of the circle.
Therefore, radius of the circle will be half of room, which is equal to rule two.
And we know area of a certainly spire square that spy into room to square, which gives you two pi,
which is option B.
Let's read the next question.
Two sides of a triangle of six centimetre and five centimeter, the third side length is in between.
All right.
Now, what's the answer to this question?
The correct answer is option E, that's between one centimeter and 11 centimeter, because we know that
some of two sites as to be greater than the third side.
Right.
So six plus five, which is equal to 11, has to be greater than the third side.
But we know the third side is less than one centimeter and difference of two sides of a triangle that's
six minus five, which is equal to one as to be less than the third side.
The one is less than the third side.
So the third side has to be between one centimeter and one centimeter.
And note, it cannot be 11 or one.
It's between them.
All right.
So that's something that you need to note.
So the answer to this question is option A.
Let's read the next question, there is a gigantic tree.
Dawn is moving away from the tree.
It takes him two minutes to change the angle of elevation from 60 degrees to 30 degrees.
The height of the tree is 30 meters behind his speed.
All right.
What's the answer to this question?
The answer to this question is option B. That square root of three by six meters per second already
over here, we have drawn a rough diagram.
This is the tree, which is 30 meters in height.
Now, initially, he was over here and the angle of elevation was 60 degrees.
Now he's moving away from the tree.
And after two minutes, his angle of elevation is 30 degrees.
All right.
Now over here and 60 will be 30 Bayway, 20 days opposite side by side.
So 10 sixty's, 30 by white and we note and 60 023, 30 by Y is equal to Rotary or Y is equal to 30
barratry, which is equal to Bend Road three.
All right.
Now ten 30 will be 30 by X plus white.
And we know 10 is equal to one barratry.
Right.
The one barratry is equal to 30 by X plus white and cross multiplying.
You get X plus why is equal to 30 Enduro three and we already know why is standard three.
Therefore X has to be thirty three minus two and three, which is equal to twenty three.
All right.
Now we need to find the speed with which he moved.
Speed is equal or distance by time.
The distance that he moved is X which is equal to twenty three meters.
Right.
And all the answer options are in meters per second.
Right.
But we need to convert two minutes into seconds or two minutes is two in the sixty seconds to the speed
is twenty three by two into sixty which is equal to room three by six meters per second, which is option
Deora here.
Let's read the next question, a cuboid and a cube are placed on a table, the dimensions of the cuboid
are three in the four and five, and that of the cube is for India, for India, for wine.
The square of the ratio of the length of the body, diagonal of the cuboid to that of the cube.
So we need to find the body diagonal dimension of both and then finding the ratio and squared.
All right.
What's the answer to this question?
It's option B. That's twenty five by twenty four.
Body diagonal of the cuboid will be root of three squared plus four square plus five square.
That's root of fifty.
Right.
And more diagonal of the cube will be root of four squared plus four squared plus four square.
That's root forty eight.
Right now if you square this and take the ratio you'll get fifty by forty eight.
That is equal to twenty five by twenty four which is option B.
All right, let's read the next question, of course, is equal to a ban is equal to one by be fined,
one by a square.
Minus one might be square.
This is what we need to find out.
But what's the answer to this question?
The answer to this question is option C, that is one.
All right.
Let's try to do this now.
We have two methods over here.
This method over here, you just need to substitute the angle, if you take it as a forty five degree
course is equal to one by two, right.
That's it.
And done is done.
Forty five.
That will be one to one baby.
So if you take one by a square, right.
That would be route to square, which is equal to two.
Right.
So this is one by a square and one by B square will be one square which is one itself.
Right.
So this is one by square, so one by a square minus.
One by B square is two minus one.
But this one itself.
All right.
And that gives you see as the answer or all the other options are eliminated.
Now, another way of doing this is one by a square is equal to one because square red square and one
by B Square is the same as Stand Square because Danny's one baby.
Now, we just need to simplify this.
We know that ten is equal to sign a vacancy.
So I can rewrite this one because square a minus sine square here, because right now over here, this
is one minus sine square eight because square and one minus nine square.
It is the same as square.
And that becomes scary because square, which is equal to one.
Now let's see why one minus nine square is equal to Cost Square.
So over here I have a triangle with A, B and C and let this angle be it.
All right.
So one minus sine square is one minus E, square by C square because sign is NBC, right?
So this is C square minus a square by square right now.
C square minus E square is equal to be square because this is the right angle triangle.
And as Pythagoras's theorem C squared as equals A squared plus B square or C square minus square will
be B square.
So I can write this as B square by square and B by C is cause of it.
So we score by C Square square.
If we are we have seen Y one minus sine square is equally square.
We use that over here and we get square by square, which is also equal to one.
And therefore the answer to this question is option C, which is one.
Let's read the next question, a circle of radius, 15 centimeter circumscribes and little triangle
in the area of the triangle.
All right.
What's the answer to this question?
The answer to this question is option E write that six hundred seventy five square root of three by
four centimeter square root.
So let's see why we get this answer.
Now, let the side of the triangle be now.
The Sock'Em Radius will be two by three, the height of the equilateral triangle.
Now the height of an equal triangle is Route three by to the side of the triangle.
But these two things we have discussed in previous videos, well, this should be very thorough.
If you have any confusion in this area, try to revisit those sections where we discussed equilateral
triangles.
All right.
Now, over here are is equal to two by three in room, three by two side.
Right.
And this, too.
And this to cancel out.
And this can be simplified aside by rule three, that is equal to fifteen centimetre because it's given
over here already.
So that means is equal to fifteen row three.
Right.
And the area of equal triangle is given us room three by four is square.
Where is the side.
That's three by four into fifteen square which is to twenty five into row three square which is three.
So that gives you two, twenty five in the three which is six seventy five.
Route three by four as the answer.
That is option eight.
Let's read the next question to cyclists moved from the same point at an angle of one thirty five degrees
and five kilometers per hour and six kilometers per hour, find the distance between them after one
hour.
All right.
What's the answer to this question?
The answer to this question is option C, that is square root of sixty one plus thirty two kilometers.
All right, let's see how to do this over here.
Let this be the starting point.
And one person is moving in this direction and the other person is moving in this direction.
And the angle between them is giving us one thirty five degrees already.
Now after one hour the person is moving in this direction, will have moved five kilometers and this
person will have moved six kilometers because that's their speed.
All right.
Now we need to find the value of that.
That is a B. That's what we need to find out.
All right.
Now, let's draw a perpendicular B.C. such that BKS perpendicular to or if this is all in BSES, perpendicular
to it.
All right.
Now, this angle is one thirty five.
So this angle will be 180, minus one thirty five, which is equal to forty five degree.
And this angle also will be equal to four different degrees because these two angles are vertically
opposite angles.
All right.
Now let's observe a triangle or triangle or B, C you can see that X divided by six is equal to sign
six sine forty five.
Right sine forty five is opposite side.
That's X by hypotenuse, that's six X by six sine forty five, which is equal to one by two.
We know sine forty five is one variable, therefore X is equal to six, Bierut two which is equal to
zero two.
But we know that X is equal to three zero two.
Now over here you can see that this angle is ninety degrees, this is forty five degrees.
So this also will be equal to forty five degrees.
Right as per angles on property of a triangle.
Now therefore these two sides have to be equal.
Therefore Y also is equal to three to this side is also equal to three to two kilometers.
Now in triangle ABC, ABC, this angle is ninety degrees so we can use Pythagoras Theorem.
So AB Square will be equal to X squared plus KBE Square.
Now AC is equal to five plus or C that's five plus three or two.
AC Square is five plus three to the whole squared plus CBS, CBS three or two.
We take three to the square.
Now let's expand this.
This is twenty five plus eighteen plus thirty root.
To impress me the whole square root is equal to A squared plus B squared plus two A B.
All right, that's twenty five plus 18, plus 30, thirty two plus 18, three to the whole square is
nine and two, that's 18 and this simplifies to 61 plus thirty two.
That's a big square.
We need to find B, so AB is equal to square root of sixty one plus thirty two, which is option C.