# The Triangle and its Properties

**Exercise 1**

**1. ** In A PQR, D is the mid-point of

PD is _____

Is QM = MR ?

**2. **Draw rough sketches for the following:

In A ABC, BE is a median.

In A PQR, PQ and PR are altitudes of the triangle.

In A XYZ, YL is an altitude in the exterior of the triangle.

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

**Exercise 2**

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

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

**Exercise 3**

**6. ** Find the value of unknown x in the following diagrams:

**7. ** Find the values of the unknowns x and y in the following diagrams:

**Exercise 4**

**8. **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

**9.** Take any point 0 in the interior of a triangle PQR. Is:

(i) OP + OQ > PQ ?

(ii) OQ + OR > QR ?

(iii) OR + OP > RP ? Since sum of two sides is greater than third side.

**10.**AM is a median of a triangle ABC. Is AB + BC + CA > 2AM?

(Consider the sides of triangles Δ ABM and Δ AMC.)

**11. **ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?

**12. ** ABCD is quadrilateral. Is AB + BC + CD + DA < 2 (AC + BD)?

**13.** 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?

**Exercise 5**

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

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

**16. ** 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.

**17. **Which of the following can be the sides of a right triangle?

(i) 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.

**18.**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.

**19. ** Angles Q and R of a Δ PQR are 25° and 65°. Write which of the following is true:

(i) PQ^{2} + QR^{2} = RP^{2}

(ii) PQ^{2} + RP^{2} = QR^{2}

(iii) RP^{2} + QR^{2} = PQ^{2}

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

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