Exercise 6.5 of Class 10 Maths Chapter 6 – Triangles is dedicated to applying the concepts of triangle similarity in real-world problems involving measurements, such as finding heights and distances. The questions are based on practical uses of the Pythagoras Theorem, converse of similarity criteria, and properties of right-angled triangles. This exercise enhances analytical thinking and geometric reasoning. The NCERT Book answers provided here follow a step-by-step format to help students clearly understand each concept and apply it confidently in board exams.
Class 10 Math Chapter 6 Triangles Ex 6.5 NCERT Textbook Answers
Ex 6.5 Class 10 Maths Question 1.
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
Solution:
Ex 6.5 Class 10 Maths Question 2.
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM X MR.
Solution:
Ex 6.5 Class 10 Maths Question 3.
In the given figure, ABD is a triangle right angled at A and AC i. BD. Show that
(i) AB2 = BC.BD
(ii) AC2 = BC.DC
(iii) AD2 = BD.CD
Solution:
Ex 6.5 Class 10 Maths Question 4.
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
Solution:
Ex 6.5 Class 10 Maths Question 5.
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2 , Prove that ABC is a right triangle.
Solution:
Ex 6.5 Class 10 Maths Question 6.
ABC is an equilateral triangle of side la. Find each of its altitudes.
Solution:
Ex 6.5 Class 10 Maths Question 7.
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
Solution:
Ex 6.5 Class 10 Maths Question 8.
In the given figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that
(i) OA2 + OB2 + OC2 – OD2 – OE2 – OF2 = AF2 + BD2 + CE2
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2.
Solution:
Ex 6.5 Class 10 Maths Question 9.
A ladder 10 m long reaches a window 8 m above the ground. ind the distance of the foot of the ladder from base of the wall.
Solution:
Ex 6.5 Class 10 Maths Question 10.
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?
Solution:
Ex 6.5 Class 10 Maths Question 11.
An aeroplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km per hour. How far apart will be the two planes after 3/2 hours?
Solution:
Ex 6.5 Class 10 Maths Question 12.
Two poles of heights 6 m and 11m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
Solution:
Ex 6.5 Class 10 Maths Question 13.
D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE2 + BD2 = AB2 + DE2.
Solution:
Ex 6.5 Class 10 Maths Question 14.
The perpendicular from A on side BC of a ∆ABC intersects BC at D such that DB = 3CD (see the figure). Prove that 2AB2 = 2AC2 + BC2.
Solution:
Ex 6.5 Class 10 Maths Question 15.
In an equilateral triangle ABC, D is a point on side BC, such that BD = 13BC. Prove that 9AD2 = 7AB2.
Solution:
Ex 6.5 Class 10 Maths Question 16.
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.
Solution:
Ex 6.5 Class 10 Maths Question 17.
Tick the correct answer and justify : In ∆ABC, AB = cm, AC = 12 cm and BC = 6 cm. The angle B is:
(a) 120°
(b) 60°
(c) 90°
(d) 45
Solution:
You can access the official NCERT Solutions for Class 10 Mathematics on the NCERT website at the following link:
NCERT Class 10 Mathematics Solutions
📘 Math & Science Solutions by Class
🔹 Class 10
🔹 Class 9
🔹 Class 8
🔹 Class 7
🔹 Class 6
🔹 Class 12
🔹 Class 11
- Class 11 Math Solutions
- Class 11 Physics Solutions
- Class 11 Chemistry Solutions
- Class 11 Biology Solutions
Class-wise Solutions
Class 12:
Class 12 Physics – NCERT Solutions
Class 12 Chemistry – NCERT Solutions
Class 11:
- Class 11 Physics – NCERT Solutions
- Class 11 Chemistry – NCERT Solutions
- Class 11 Biology – NCERT Solutions
- Class 11 Math – NCERT Solutions
Class 10:
Class 9:
Class 8:
Class 7:
Class 6:
Subject-wise Solutions
Physics:
Chemistry:
Biology:
Math:
- Class 11 Math – NCERT Solutions
- Class 10 Math – NCERT Solutions
- Class 9 Math – NCERT Solutions
- Class 8 Math – NCERT Solutions
Science:
- Class 10 Science – NCERT Solutions
- Class 9 Science – NCERT Solutions
- Class 8 Science – Oxford Solutions
- Class 7 Science – Oxford Solutions
- Class 6 Science – Oxford Solutions