NCERT Solutions for Class 9 Maths Exercise 8.1 Question 1
Welcome, students! Let’s solve the first question from the chapter on Quadrilaterals. This problem is a classic application of the angle sum property of a quadrilateral and the concept of ratios.
| Given Information | The angles of a quadrilateral are in the ratio &&3 : 5 : 9 : 13&&. |
|---|---|
| To Find | The measure of all four angles of the quadrilateral. |
| Final Answer | The angles are &&36^\circ, 60^\circ, 108^\circ,&& and &&156^\circ&&. |
Question 1: The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
How to Find the Angles of the Quadrilateral 🤔
This problem can be solved in a few straightforward steps.
- Step 1: Represent the Angles Algebraically
When quantities are given in a ratio, like &&3:5:9:13&&, represent them using a common multiplier ‘&&x&&’. So, the four angles can be written as &&3x, 5x, 9x,&& and &&13x&&. - Step 2: Apply the Angle Sum Property of a Quadrilateral
Recall the fundamental theorem that the sum of all four interior angles of any quadrilateral is always &&360^\circ&&. This property is the basis for our equation. - Step 3: Set Up and Solve the Equation for ‘&&x&&’
Create an equation by summing the algebraic representations of the angles and setting the total equal to &&360^\circ&&. The equation is: &&3x + 5x + 9x + 13x = 360^\circ&&. Combine the terms to get &&30x = 360^\circ&&. Solve for &&x&& by dividing &&360&& by &&30&&, which gives &&x = 12&&. - Step 4: Calculate Each Individual Angle
Substitute the value of &&x (12)&& back into the expression for each angle:- First angle = &&3 \times 12^\circ = 36^\circ&&
- Second angle = &&5 \times 12^\circ = 60^\circ&&
- Third angle = &&9 \times 12^\circ = 108^\circ&&
- Fourth angle = &&13 \times 12^\circ = 156^\circ&&
- Step 5: Verify the Solution
As a final check, add the calculated angles to ensure they sum to &&360^\circ&&. &&36^\circ + 60^\circ + 108^\circ + 156^\circ = 360^\circ&&. The solution is correct.
Detailed Step-by-Step Solution 📝
Let the four angles of the quadrilateral be &&\angle A, \angle B, \angle C,&& and &&\angle D&&.
Given: The ratio of the angles is &&\angle A : \angle B : \angle C : \angle D = 3 : 5 : 9 : 13&&.
Let the common ratio be &&x&&. Then, we can express the angles as:
- &&\angle A = 3x&&
- &&\angle B = 5x&&
- &&\angle C = 9x&&
- &&\angle D = 13x&&
We know that by the Angle Sum Property of a Quadrilateral, the sum of all its interior angles is &&360^\circ&&.
&&\implies \angle A + \angle B + \angle C + \angle D = 360^\circ&&
Now, substitute the values in terms of &&x&&:
&&3x + 5x + 9x + 13x = 360^\circ&&
Combine the terms:
&&30x = 360^\circ&&
Solve for &&x&&:
&&x = \frac{360^\circ}{30} = 12^\circ&&
Now that we have the value of &&x&&, we can find the measure of each angle:
- &&\angle A = 3x = 3 \times 12^\circ = 36^\circ&&
- &&\angle B = 5x = 5 \times 12^\circ = 60^\circ&&
- &&\angle C = 9x = 9 \times 12^\circ = 108^\circ&&
- &&\angle D = 13x = 13 \times 12^\circ = 156^\circ&&
Conclusion:
The four angles of the quadrilateral are &&36^\circ, 60^\circ, 108^\circ,&& and &&156^\circ&&.
Key Strategy for Ratio Problems ✅
Whenever you see quantities (like angles or side lengths) given in a ratio, such as &&a : b : c&&, the best first step is always to introduce a common multiplier, &&x&&. This allows you to write the quantities as &&ax, bx,&& and &&cx&&, turning the ratio into a simple algebraic equation that you can solve.
FAQ (Frequently Asked Questions)
Q: What is the Angle Sum Property of a Quadrilateral?
A: The Angle Sum Property of a Quadrilateral states that the sum of the four interior angles of any convex quadrilateral is always &&360&& degrees.
Q: How is the Angle Sum Property of a Quadrilateral derived?
A: You can derive this property by drawing a diagonal across the quadrilateral. The diagonal splits the quadrilateral into two triangles. Since the sum of angles in each triangle is &&180^\circ&&, the total sum of angles in the quadrilateral is &&180^\circ + 180^\circ = 360^\circ&&.
Q: How should I approach any problem where quantities are given in a ratio?
A: The standard method is to introduce a common variable or multiplier (usually ‘&&x&&’). If the ratio is &&a:b:c&&, you would represent the quantities as &&ax, bx,&& and &&cx&&. This allows you to form an algebraic equation based on other information given in the problem, such as their sum.
Q: What is the sum of the ratio parts in this question?
A: The sum of the ratio parts is &&3 + 5 + 9 + 13 = 30&&. You can find the value of one ‘part’ (&&x&&) by dividing the total sum of angles (&&360^\circ&&) by this sum of ratio parts (&&30&&), which gives &&12&&.
