M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 12 pythagoras theorem

Lengths of 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) 3 cm, 8 cm, 6 cm

(ii) 13 cm, 12 cm, 5 cm

(iii) 1.4 cm, 4.8 cm, 5 cm

Foot of a 10 m long ladder leaning against a vertical well is 6 m away from the base of the wall. Find the

height of the point on the wall where the top of the ladder reaches.

In a right-angled triangle, if hypotenuse is 20 cm and the ratio of the other two sides is 4:3, find the sides.

For going to a city B from city A, there is a route via city C such that AC ⊥ CB, AC = 2x km and CB=2(x+

7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find

how much distance will be saved in reaching city B from city A after the construction of the highway.

ABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. Find the altitude on BC and Hence,

calculate its area.

(a) In figure (i) given below, AB = 12 cm, AC = 13 cm, CE = 10 cm and DE = 6 cm. Calculate the length

of BD.

(b) In figure (ii) given below, ∠PSR = 90°, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of

PR.

(c) In figure (iii) given below, ∠ D = 90°, AB = 16 cm, BC = 12 cm and CA = 6 cm. Find CD.

(a) In figure (i) given below, BC = 5 cm,

∠B =90°, AB = 5AE, CD = 2AE and AC = ED. Calculate the lengths of EA, CD, AB and AC.

(b) In the figure (ii) given below, ABC is a right triangle right angled at C. If D is mid-point of BC, prove

that \mathbf{A B}^{2}=4 \mathbf{A} \mathbf{D}^{2}-3 \mathbf{A} \mathbf{C}^{2}

In a rhombus, If diagonals are 30 cm and 40 cm, find its perimeter.

a) In figure (i) given below, AB || DC, BC = AD = 13 cm. AB = 22 cm and DC = 12cm. Calculate the

height of the trapezium ABCD.

(b) In figure (ii) given below, AB || DC, ∠ A = 90°, DC = 7 cm, AB = 17 cm and AC = 25 cm. Calculate BC.

(c) In figure (iii) given below, ABCD is a square of side 7 cm. if

AE = FC = CG = HA = 3 cm,

(i) prove that EFGH is a rectangle.

(ii) find the area and perimeter of EFGH.

AD is perpendicular to the side BC of an equilateral Δ ABC. Prove that 4 \mathrm{AD}^{2}=3 \mathrm{AB}^{2}

If AD, BE and CF are medians of

\Delta \mathrm{ABC}, \text { prove that } 3\left(A B^{2}+B C^{2}+C A^{2}\right)=4\left(A D^{2}+B E^{2}+C F^{2}\right)

In a quadrilateral ABCD, \angle \mathrm{B}=90^{\circ}=\angle \mathrm{D} . \text { Prove that } 2 \mathrm{AC}^{2}-\mathrm{BC}^{2}=\mathrm{AB}^{2}+\mathrm{AD}^{2}+\mathrm{DC}^{2}