The diameter of the tyres of a car is 63 centimetres. If the car is driven at a constant speed such that the tyre makes one 360-degree rotation in 0.24 seconds, what is the approximate speed of the car in kilometres per hour?
Given:
To find: Approx. speed of car in kilometers per hour ( in short kmph)
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
Point A lies on a circle whose center is at point C. Does point B lie inside the circle?
Steps 1 & 2: Understand Question and Draw Inferences
Given: The given information can be represented visually as in the figure above
To find: Does point B lie inside the circle?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘BC^{2} = AC^{2} + AB^{2}’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘∠CAB is greater than ∠ABC’
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of Steps 3 and 4, this step is not required
Answer: Option D
The tyre of a car made an average of 500 revolutions per minute in a particular trip. If the radius of the tyre was 10 centimeters and the car was driven for 1 hour without any stoppages, approximately how much distance in kilometers was covered in the trip?
Given
To Find: Total distance covered by the car in kilometers?
Approach
After calculating the total distance covered, we need to convert it into kilometers
Working Out
Answer: C
ΔABC, which is right-angled at B, is inscribed in a circle with centre O and radius 6 units. If the length of the smaller arc between points A and B is 4π units, what is the length of line segment BC?
Given:
To find: x = ?
Approach and Working:
Looking at the answer choices, we see that the correct answer is Option D.
In the figure above, triangle ABC is inscribed in the circle with center O, such that CD is perpendicular to AB. If the length of the side AC is centimeters, ∠AOB is 60^{o} and the smaller perimeter of the sector AOB is 12 + 2π centimeters, what is the perimeter of the triangle ABC in centimeters?
Given
To Find: Perimeter of triangle ABC
Approach
Working Out
hence BA = 6 centimeters
In the given figure, AB and CD are the longest chords of the circle with their lengths equal to 8 units. If the length of the minor arc BC is 1/6th of the perimeter of the circle, what is the length of the chord BD?
Given
To Find: Length of chord BD?
Approach
Now as triangle OED is a right angled triangle, knowing one of the sides and one of the angles will be sufficient to calculate the length of ED.
Working Out
1. Length of arc which gives us ∠COB = 60^{o}
4. As triangle ODE is a 30^{o} - 60^{o} -90^{o} triangle, we have the length of ED = 2√3
5. Thus BD = 2 * 2√3 = 4√3
Answer: C
If O is the center of the circle above and the length of chord AB is 2 units, what is the length of the arc ACB?
(1) The area of ΔOAB is √3 square units
(2) The area of sector OACBO is 2π/3
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Step 3: Analyze Statement 1 independently
(1) Statement 1 states that "The area of ΔOAB is √3 square units."
r=2 (Rejecting the negative root since radius cannot be negative))
Step 4: Analyze Statement 2 independently
(2) Statement 2 states that "The area of sector OACBO is 2π/3
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 3, this step is not required
Hence the correct answer is Option A .
The annual money spent by a company on search engine marketing, content marketing and affiliate marketing is to be shown on a circlular graph. If the size of each sector of the graph is proportional to the amount of money it represents, did the company spend the most money on search engine marketing?
(1) The angle of the sector that represents content marketing is 80^{o}
(2) The angle of the sector that represents affiliate marketing is 210^{o}
Steps 1 & 2: Understand Question and Draw Inferences
Given: 3 amounts to be represented on a circle graph:
To find: Was money spent on Search Engine Marketing the greatest?
^{}
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The angle of the sector that represents content marketing is 80∘
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The angle of the sector that represents affiliate marketing is 210∘
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
The figure above shows a rectangular plot of land OPQR. In this plot, a patch of land OABO, where the path AB represents a circular arc of radius 20√2 metres, is developed as a lawn. The point A is at a distance of 20 metres each from sides OR and OP. The point B is at a distance of 10√6 metres from side OP and 10√2 metres from side OR. What is the area, in square metres, of the intended lawn?
Given:
To find: Area of region OABO
Approach:
Working Out:
Thus, we see that OA = OB = Radius of the circular arc AB =20√2 meters
Thus, ∠BOA = 15^{o}
Finding the area of sector OABO
Looking at the answer choices, we see that the correct answer is Option B
A circle is inscribed in a square ABCD such that the circle touches all the sides of the square. If the perimeter of the shaded region is 24 + 6π, what is the area of the shaded region?
Given
A circle is inscribed in the square ABCD
Let the side of square be x and the radius of the circle be r
To Find: Area of the shaded region?
Area of the shaded region = (3over 4)∗(Area of square – Area of circle)
Out of these 4 parts:
Approach
3. Since area of square and area of circle are in terms of r, we need to find the value of r, by equating perimeter of shaded region to 24 + 6π.
Perimeter of the shaded region = Perimeter of the square + Perimeter of the circle – perimeter of the region AFE
Working Out
2. Area of shaded region = 3/4 ∗(Area of square – Area of circle)
Two concentric circles have their centers at point O such that a line segment AB having its end points on the outer circle touches the inner circle at point C. The length of the line segment AB is times the radius of the inner circle. If an equilateral triangle is drawn such that the area of the triangle is equal to the ratio of the radii of the outer circle and the inner circle respectively, what is the length of the side of the triangle?
Given
To Find:
Approach
So, OC will be perpendicular to AB.
Working Out
(rejecting the negative root since the ratio of radii cannot be negative)
2.
Therefore, a=2 , as a being the length of a triangle, cannot be negative.
Hence the correct answer is Option C .
If a circle has the same area as right triangle PQR shown in the figure above. Which of the following is the closest in value to the radius of the circle?
Given:
To find: r = ?
Approach:
b. Therefore, to find r, we need to find A
Working Out:
c. The value of will be very slightly greater than 1
d. We know that Since 7 is slightly greater than the mid-point between 4 and 9 (which is
the value of √7 will be slightly greater than the mid-point between So, the value of √7 will be slightly greater than 2.5
e. So, the product of and √7 will be slightly greater than 2.5
f. Therefore, the closest integer to the value of r will be 3
Answer: Option A
In the figure above, triangle ABC is inscribed in a circle whose centre O has the x- and y-coordinates as (0,0). If the x- and y- coordinates of point A are (-4,0) and ∠BAC = 30^{∘}, what is the area, in square units, of triangle AOB?
Given:
O is the center of the circle and A(-4,0) is a point on the circle
So, ∠ABC is an angle in the semicircle
So, by Angle Sum Property, ∠ACB = 60^{o}
To find:
Approach:
Area triangle AOB = ½ * AO * BD
So, this is an isosceles triangle.
OD: BD: OB = 1: √3: 2
Lines CA and CB touch the circle only at points A and B respectively, as shown in the figure above. If the centre O of the circle is at the point (0,0), the coordinates of point C are (0,2) and the radius of the circle is √2 units, what are the coordinates of point A?
Given:
To find:
Approach and Working:
In the coordinate system, the center of a circle lies at (2, 3). If point A with coordinates (-1, 7) does not lie outside the circle, which of the following points must lie inside the circle?
I. (0, 7)
II. (5, -1)
III. (-2, 7)
Given
To Find: Which of the points in the options must lie inside the circle?
Approach
In general, it can be said that the distance between any two points is =
So we can find the distance between Point A and the centre of the circle by the above formula
Working Out
Hence, we see that only option I (0, 7) always lie inside the circle.
Answer: A
15 children are given tags numbered from 1 to 15 and are seated in a circular formation in the increasing order of their respective tag numbers. The total area covered by the circular formation is 36π square units and the distance between any two neighbouring children in the formation is equal., If the number of children seated between the child with tag number m and the child with tag number 1 is equal to the number of children seated between the child with tag number m and the child with tag number 15, what is the minimum distance covered along the circular formation by the child with tag number 1 to reach the child with tag number m -2 and then go back to his original position?
Given
To Find:
Approach
Working Out
The figure above shows a circle whose centre O has the x- and y- coordinates as (6,0). Points A(m,n) and B(8,2) are marked on the circle such that ∠AOB = 105^{∘}. If the circle is symmetrical about the x-axis, what is the value of m?
Given:
To find: m = ?
Approach:
From this diagram, it’s clear that there are 2 ways to find the value of m:
2. We are given the coordinates of both points O and B. So, using these, we can find the radius OB = OA
3. Also, if we drop a perpendicular from point B on the x-axis, we’ll know the magnitude of all the sides of the resulting right triangle.
So, we can use trigonometric ratios to find ∠BOE.
So, we can find ∠AOF using the equation
(Thus, in this Approach, we’re using Way 1 to find m)
Working Out:
Looking at the answer choices, we see that the correct answer is Option D
If O is the center of the circle above, what fraction of the circular region is encompassed by an angle of x degrees?
In this question, the angles shown appear span half the circle. We can infer that they must, in fact, because the opposite angles of bisecting lines are equal. On the other side of each of the 2x angles shown is another 2x, and on the other side of each x shown is another x. Therefore,
2x+x+2x+x = 180
half the span of the circle. Thus, and . Since the full span of a circle is 360 degrees, the angle x spans a fraction of 30/360 = 1/12
The correct answer is (B).
In the coordinate plane, a circle has center (-1, -3) and passes through the point (4,2). What is the circumference of the circle?
To find the circumference of this circle, we need its radius. And the radius is the distance from its center to any point on the circle, so the radius is the distance from (-1 ,-3) to (4,2).
This distance is computed with the distance formula, which is essentially the Pythagorean Theorem. The sides of the triangle are the x and y distances, 5 and 5, so the distance is
The radius of the circle is 5√2, so the circumference is
The correct answer is (B).
The circle with center C shown above is tangent to both axes and has an area of A. What is the distance from O to C, in terms of A?
In this question, we can find the radius r in terms of A by manipulating the area formula, A= πr^{2}:
Point C in the figure is the upper-right point of a square that has sides of length r. And the distance from the origin to C is the diagonal of the square, so it is the hypotenuse of a right triangle with height r and base r, so the distance will equal
The correct answer is (B).
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