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 ⊥ QR. Show that PM2 = QM X MR.
In a right triangle, perpendicular drawn from right angle to hypotenuse divides the triangle into two similar triangles
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
ABC is an isosceles triangle right angled at C. Prove that AB2 = 2AC2.
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, Prove that ABC is a right triangle.
ABC is an equilateral triangle of side la. Find each of its altitudes.
As all the altitudes of an equilateral triangle are equal hence, each of the altitudes of ∆ABC is of length .
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
The figure given below shows a rhombus ABCD in which AB = BC = CD = DA. The diagonals AC and BD bisect each other at O.
In ∆AOB, ∠AOB = 90°
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.
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.
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 airplane leaves an airport and flies due north at a speed of 1000 km per hour. At the same time, another airplane 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 1 hours?
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.
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.
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.
In an equilateral triangle ABC, D is a point on side BC, such that BD = BC. Prove that 9AD2 = 7AB2.
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 ∆ABC, AB = 6cm, AC = 12 cm and BC = 6 cm. The angle B is: