Premium Online Home Tutors

3 Tutor System

Starting just at 265/hour

Answer :

Given, the perpendicular from A on side BC of a \(\triangle\) ABC intersects BC at D such that;

DB = 3CD.

In \(\triangle\) ABC,

AD \(\perp\) BC and BD = 3CD

In right angle triangle, ADB and ADC, by Pythagoras theorem,

\(AB^2 = AD^2 + BD^2\) ……………………….(i)

\(AC^2 = AD^2 + DC^2\) ……………………………..(ii)

Subtracting equation (ii) from equation (i), we get

\(AB^2 – AC^2 = BD^2 – DC^2\)

\(= 9CD^2 – CD^2\) [\(\because \) BD = 3CD]

\(= 9CD^2 = 8(BC/4)^2 \)

[\(\because \) BC = DB + CD = 3CD + CD = 4CD]

\(\therefore AB^2 – AC^2 = {{BC^2} \over {2}}\)

\(\Rightarrow 2(AB^2 – AC^2) = BC^2\)

\( \Rightarrow 2AB^2 – 2AC^2 = BC^2\)

\(\Rightarrow 2AB^2 = 2AC^2 + BC^2\).

- Sides of triangles are given below. Determine which of them are right triangles? In case of a right triangle, write the length of its hypotenuse. (i) 7 cm, 24 cm, 25 cm (ii) 3 cm, 8 cm, 6 cm (iii) 50 cm, 80 cm, 100 cm (iv) 13 cm, 12 cm, 5 cm
- PQR is a triangle right angled at P and M is a point on QR such that PM \(\perp\) QR. Show that \(PM^2 = QM × MR\).
- In Figure, ABD is a triangle right angled at A and AC \(\perp\) BD. Show that (i) \(AB^2 = BC × BD\) (ii) \(AC^2 = BC × DC\) (iii) \(AD^2 = BD × CD\)
- ABC is an isosceles triangle right angled at C. Prove that \(AB^2 = 2AC^2\).
- ABC is an isosceles triangle with AC = BC. If \(AB^2 = 2AC^2\), prove that ABC is a right triangle.
- ABC is an equilateral triangle of side 2a. Find each of its altitudes.
- Prove that the sum of the squares of the sides of rhombus is equal to the sum of the squares of its diagonals.
- In Fig. 6.54, O is a point in the interior of a triangle. ABC, OD \(\perp\) BC, OE \(\perp\) AC and OF \(\perp\) AB. Show that: \( (i) OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2 = AF^2 + BD^2 + CE^2\) , \( (ii) AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2\).
- A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from base of the wall.
- A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
- An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after \({1} {\dfrac{1}{2}}\) hours?
- Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
- D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that \(AE^2 + BD^2 = AB^2 + DE^2\).
- In an equilateral triangle ABC, D is a point on side BC such that BD = \(\frac{1}{3} \) BC. Prove that \(9AD^2 = 7AB^2\).
- In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
- Tick the correct answer and justify: In \(\triangle\) ABC, AB = \(6 \sqrt {3}\) cm, AC = 12 cm and BC = 6 cm. The angle B is: (A) 120° (B) 60° (C) 90° (D) 45°

- NCERT solutions for class 10 maths chapter 1 Real Numbers
- NCERT solutions for class 10 maths chapter 1 Electricity, Light , Carbon and it's compounds
- NCERT solutions for class 10 maths chapter 2 Polynomials
- NCERT solutions for class 10 maths chapter 3 Pair of linear equations in two variables
- NCERT solutions for class 10 maths chapter 4 Quadratic Equations
- NCERT solutions for class 10 maths chapter 5 Arithmetic Progressions
- NCERT solutions for class 10 maths chapter 6 Triangles
- NCERT solutions for class 10 maths chapter 7 Coordinate Geometry
- NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry
- NCERT solutions for class 10 maths chapter 9 Some Applications of Trigonometry
- NCERT solutions for class 10 maths chapter 10 Circles
- NCERT solutions for class 10 maths chapter 11 Constructions
- NCERT solutions for class 10 maths chapter 12 Areas related to circles
- NCERT solutions for class 10 maths chapter 13 Surface Areas and Volumes
- NCERT solutions for class 10 maths chapter 14 Statistics
- NCERT solutions for class 10 maths chapter 15 Probability

- NCERT solutions for class 10 science chapter 1 Chemical Reactions and Equations
- NCERT solutions for class 10 science chapter 2 Acids, Bases and Salts
- NCERT solutions for class 10 science chapter 3 Metals and Non Metals
- NCERT solutions for class 10 science chapter 4 Carbon and its Compounds
- NCERT solutions for class 10 science chapter 5 Periodic Classification of Elements
- NCERT solutions for class 10 science chapter 6 Life Processes
- NCERT solutions for class 10 science chapter 7 Control and Coordination
- NCERT solutions for class 10 science chapter 8 How do Organisms Reproduce
- NCERT solutions for class 10 science chapter 9 Heredity and Evolution
- NCERT solutions for class 10 science chapter 10 Light Reflection and Refraction
- NCERT solutions for class 10 science chapter 11 Human Eye and Colorful World
- NCERT solutions for class 10 science chapter 12 Electricity
- NCERT solutions for class 10 science chapter 13 Magnetic Effect of Electric Current
- NCERT solutions for class 10 science chapter 14 Sources of Energy
- NCERT solutions for class 10 science chapter 15 Our Environment
- NCERT solutions for class 10 science chapter 16 Management of Natural Resources