Find the best tutors and institutes for Class 10 Tuition
Q5:
The altitude of a right triangle is $7$ cm less than its base. If the hypotenuse is $13$ cm, find the other two sides.
Solution :
Given:
- A right-angled triangle.
- The length of the hypotenuse is $13$ cm.
- The altitude (height) is $7$ cm less than the base.
To find:
- The lengths of the base and the altitude of the triangle.
Step 1: Defining Variables
Let the base of the right triangle be $x$ cm.
According to the problem, the altitude is $7$ cm less than the base.
Therefore, the altitude = $(x - 7)$ cm.
Step 2: Applying the Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagorean Theorem):
$(\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Altitude})^2$
Substituting the given values:
$13^2 = x^2 + (x - 7)^2$
Step 3: Expanding and Simplifying the Equation
$169 = x^2 + (x^2 - 14x + 49)$ [Using the identity $(a - b)^2 = a^2 - 2ab + b^2$]
$169 = 2x^2 - 14x + 49$
Subtract $169$ from both sides to set the quadratic equation to zero:
$0 = 2x^2 - 14x + 49 - 169$
$2x^2 - 14x - 120 = 0$
Divide the entire equation by $2$ to simplify:
$x^2 - 7x - 60 = 0$
Step 4: Solving the Quadratic Equation by Factorization
We need to find two numbers that multiply to $-60$ and add to $-7$. These numbers are $-12$ and $5$.
$x^2 - 12x + 5x - 60 = 0$
$x(x - 12) + 5(x - 12) = 0$
$(x - 12)(x + 5) = 0$
Step 5: Determining the Possible Values for x
Setting each factor to zero:
1) $x - 12 = 0 \implies x = 12$
2) $x + 5 = 0 \implies x = -5$
Since the length of a side of a triangle cannot be negative, we discard $x = -5$.
Therefore, the base $x = 12$ cm.
Step 6: Calculating the Altitude
Altitude = $x - 7$
Altitude = $12 - 7 = 5$ cm.
Final Answer: The base of the triangle is 12 cm and the altitude is 5 cm.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.2
- Q1(i): Find the roots of the following quadratic equations by factorisation: (i) $x^2 – 3x – 10 = 0$
- Q1(ii): Find the roots of the following quadratic equations by factorisation: (ii) $2x^2 + x – 6 = 0$
- Q1(iii): Find the roots of the following quadratic equations by factorisation: (iii) $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$
- Q1(iv): Find the roots of the following quadratic equations by factorisation: (iv) $2x^2 – x + \frac{1}{8} = 0$
- Q1(v): Find the roots of the following quadratic equations by factorisation: (v) $100x^2 – 20x + 1 = 0$
- Q2: Solve the problems given below. Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs.750. We would like to find out the number of toys produced on that day.
- Q3: Find two numbers whose sum is $27$ and product is $182$.
- Q4: Find two consecutive positive integers, sum of whose squares is $365$.
- Q6: A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was $3$ more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ $90$, find the number of articles produced and the cost of each article.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Quadratic Equations
Gaur City 2, Noida
5 yrs of
Exp
500
per hour
Online Classes
Student's Home
Tutor's home
I am a mathematics teacher I am giving home tution since 4 years I have done postgraduation in mathematics I am teaching in school last 4 years.
Top Tutor
Patthri Bagh, Dehradun
6 yrs of
Exp
1 student
3 hrs of classes completed
500
per hour
Online Classes
I worked as a Computer teacher in Kendriya Vidyalaya. I worked for a software company Synapse Valutech Private Limited as a Java developer. I worked for so many training centers for Java-like DUCAT, High Technologies, Sky Infotech. Special batches for freshers in JAVA and for working professionals. Special Batches for BA, XI-XII for History, Geography, Economics, Indian Polity, Environment, SSC Maths, English, GS. Vast experience in teaching for class 1 to 10( Science, Maths, SST, Hindi, English Grammar, Computer Science or IT). I have students of DPS Vasant Kunj, DAV Vasant Kunj, Mother's International, Bhatnagar International school, DU and JNU.
Carterpuri, Palam Vihar, Gurgaon
20 yrs of
Exp
500
per hour
Student's Home
I am a professional teacher with more than 25 years of teaching experience in prestigious schools of India.My teaching assures value for money and produce best results.I emphasize on concepts clarity and create natural interest in the subject.
Top Tutor
Kamal Chowk, Nagpur
20 yrs of
Exp
Tutor's home
I have 20 years of experience in the field of teaching. I conduct classes for Spoken English, BBA, MBA, Hindi Language, Career Counselling and Classes I to X. I tutor (offline/online) for syllabi-based subjects Classes I to X for CBSE, ICSE, IGCSE, IB Board. For classes X and above, I focus on tutoring subjects like Hindi, English, Psychology. Accountancy, Business Studies, Mathematics, Physics, Chemistry for CBSE, ICSE, IGCSE and IB Board. I also conduct career counselling and psychological counselling. I also assist students in pursuing MBA/PGDBM courses in their dissertation work. People engaged in doctoral theses in the domains of management and psychology engage my services for their research work. Since there is a lot of demand for enhancing one's conversational English, this is what I am focusing on at the moment. Being bilingual, my medium of communication is English as well as Hindi. I train learners in the usage of WordPress, Joomla - to develop their websites; right from domain registration to site maintenance. Besides, I assist teachers in pursuing Teachers' Training Courses.
Top Tutor
Sector 49 South City II, Gurgaon
16 yrs of
Exp
500
per hour
Online Classes
Student's Home
I am an engineer and i have been teaching for the past 16 years. I put stress on the clarification of the basic concepts of the subject. I teach maths and science.
Find more Tutor for Quadratic Equations in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
