Advertisements
Advertisements
प्रश्न
The angles of a cyclic quadrilateral ABCD are ∠A = (6x + 10)°, ∠B = (5x)°, ∠C = (x + y)°, ∠D = (3y – 10)°. Find x and y, and hence the values of the four angles.
Advertisements
उत्तर
We know that, by property of cyclic quadrilateral,
Sum of opposite angles = 180°
∠A + ∠C = (6x + 10)° + (x + y)° = 180° ......[ ∵ ∠A = (6x + 10)°, ∠C = (x + y)°, given]
⇒ 7x + y = 170° ......(i)
And ∠B + ∠D = (5x)° + (3y – 10)° = 180° .....[∵ ∠B = (5x)°, ∠D = (3y – 10)°, given]
⇒ 5x + 3y = 190°
On multiplying equation (i) by 3 and then subtracting equation (ii) from them, we get
3 × (7x + y) – (5x + 3y) = 510° – 190°
⇒ 21x + 3y – 5x – 3y = 320°
⇒ 16x = 320°
⇒ x = 20°
On putting x = 20° in equation (i), we get
7 × 20 + y = 170°
⇒ y = 170° – 140°
⇒ y = 30°
∴ ∠A = (6x + 10)°
= 6 × 20° + 10°
= 120° + 10°
= 130°
∠B = (5x)°
= 5 × 20°
= 100°
∠C = (x + y)°
= 20° + 30°
= 50°
∠D = (3y – 10)°
= 3 × 30° – 10°
= 90° – 10°
= 80°
Hence, the required values of x and y are 20° and 30°, respectively and the values of the four angles i.e., ∠A, ∠B, ∠C and ∠D are 130°, 100°, 50° and 80°, respectively.
APPEARS IN
संबंधित प्रश्न
Solve the following system of equations by using the method of elimination by equating the co-efficients.
`\frac { x }{ y } + \frac { 2y }{ 5 } + 2 = 10; \frac { 2x }{ 7 } – \frac { 5 }{ 2 } + 1 = 9`
Solve the following pair of linear equation by the elimination method and the substitution method:
3x + 4y = 10 and 2x – 2y = 2
Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Two types of boxes A, B are to be placed in a truck having a capacity of 10 tons. When 150 boxes of type A and 100 boxes of type B are loaded in the truck, it weighes 10 tons. But when 260 boxes of type A are loaded in the truck, it can still accommodate 40 boxes of type B, so that it is fully loaded. Find the weight of each type of box.
Out of 1900 km, Vishal travelled some distance by bus and some by aeroplane. The bus travels with an average speed of 60 km/hr and the average speed of the aeroplane is 700 km/hr. It takes 5 hours to complete the journey. Find the distance, Vishal travelled by bus.
The sum of a two-digit number and the number formed by reversing the order of digit is 66. If the two digits differ by 2, find the number. How many such numbers are there?
Sanjay gets fixed monthly income. Every year there is a certain increment in his salary. After 4 years, his monthly salary was Rs. 4500 and after 10 years his monthly salary became 5400 rupees, then find his original salary and yearly increment.
The sum of the digits in a two-digits number is 9. The number obtained by interchanging the digits exceeds the original number by 27. Find the two-digit number.
If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 9 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square units. Then find the length and breadth of the rectangle.
Solve the following simultaneous equation.
x − 2y = −2 ; x + 2y = 10
Solve the following simultaneous equation.
`x/3 + y/4 = 4; x/2 - y/4 = 1`
The difference between an angle and its complement is 10° find measure of the larger angle.
Difference between two numbers is 3. The sum of three times the bigger number and two times the smaller number is 19. Then find the numbers
The length of the rectangle is 5 more than twice its breadth. The perimeter of a rectangle is 52 cm, then find the length of the rectangle
The angles of a triangle are x, y and 40°. The difference between the two angles x and y is 30°. Find x and y.
Read the following passage:
Two schools 'P' and 'Q' decided to award prizes to their students for two games of Hockey ₹ x per student and Cricket ₹ y per student. School 'P' decided to award a total of ₹ 9,500 for the two games to 5 and 4 Students respectively; while school 'Q' decided to award ₹ 7,370 for the two games to 4 and 3 students respectively. |
Based on the above information, answer the following questions:
- Represent the following information algebraically (in terms of x and y).
- (a) What is the prize amount for hockey?
OR
(b) Prize amount on which game is more and by how much? - What will be the total prize amount if there are 2 students each from two games?
