The Triangle and its Properties

Δ ABC is right-angled at C. If
AC = 5 cm and BC = 12 cm find
the length of AB.

Determine whether the triangle whose lengths of sides are 3 cm, 4 cm,
5 cm is a right-angled triangle.

The lengths of two sides of a triangle are 6 cm and 8 cm. Between which
two numbers can length of the third side fall?

Is there a triangle whose sides have lengths 10.2 cm, 5.8 cm and 4.5 cm?

The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.

Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.

Angles Q and R of a ΔPQR are 25° and 65°.

Write which of the following is true:

(i) PQ2 + QR2= RP2

(ii) PQ2 + RP2= QR2

(iii) RP2 + QR2= PQ2

A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.

Which of the following can be the sides of a right triangle?

(i) 2.5 cm, 6.5 cm, 6 cm

(ii) 2 cm, 2 cm, 5 cm

(iii) 1.5 cm, 2 cm, 2.5 cm

In the case of right-angled triangles, identify the right angles.

A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.

ABC is a triangle right angled at C. If AB = 25 cm and AC = 7 cm, find BC.

PQR is a triangle right angled at P. If PQ = 10 cm and PR = 24 cm, find QR.

The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?

ABCD is quadrilateral.

Is AB + BC + CD + DA < 2 (AC + BD)?

ABCD is quadrilateral.

Is AB + BC + CD + DA >AC + BD?

AM is a median of a triangle ABC.

Is AB + BC + CA > 2 AM?

(Consider the sides of triangles ΔABM and ΔAMC.)

Take any point O in the interior of a triangle PQR. Is

(i) OP + OQ > PQ?

(ii) OQ + OR > QR?

(iii) OR + OP > RP?

Is it possible to have a triangle with the following sides?

(i) 2 cm, 3 cm, 5 cm (ii) 3 cm, 6 cm, 7 cm

(iii) 6 cm, 3 cm, 2 cm

Find the value of the unknowns x and y in the following diagrams:

Find the value of the unknown x in the following diagrams:

Find the value of the unknown interior angle x in the following figures:

Find the value of the unknown exterior angle x in the following diagrams:

Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same.

Draw rough sketches for the following:

(a) In ΔABC, BE is a median.

(b) In ΔPQR, PQ and PR are altitudes of the triangle.

(c) In ΔXYZ, YL is an altitude in the exterior of the triangle.

In ΔPQR, D is the mid-point of .

is __________.

PD is __________.

Is QM = MR?