Maths Pythagoras Theorem Exercise 6

Pythagoras Theorem Exercise 12

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Q1 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

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Q2 Foot of a 10 m long ladder leaning against a vertical well is 6 m away from the base of the wail. Find the height of the point on the wall where the top of the ladder reaches

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Q3 A guy attached a wire 24 m long to a vertical pole of height 18 m 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 taught?

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Q4 Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

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Q5 In a right-angled triangle, if hypotenuse is 20 cm and the ratio of the other two sides is 4:3, find the sides.

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Q6 If the sides of a triangle are in the ratio 3:4:5, prove that it is right-angled triangle.

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Q7 For going to a city B from city A, there is 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 highway.

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Q8 The hypotenuse of right triangle is 6m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.

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Q9 ABC is an isosceles triangle right angled at C. Prove that AB² = 2AC².

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Q10 In a triangle ABC, AD is perpendicular to BC. Prove that AB² + CD² = AC² + BD².

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Q11 In ∆PQR, PD ⊥ QR, such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b) (a - b) = (c + d) (c - d).

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Q12 ABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. Find the altitude on BC and Hence, calculate its area.

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Q13 Find the area and the perimeter of a square whose diagonal is 10 cm long.

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Q14 (a) In fig. (i) given below, ABCD is a quadrilateral in which AD = 13 cm, DC = 12 cm, BC = 3 cm, ∠ ABD = ∠BCD = 90°. Calculate the length of AB.

(b) In fig. (ii) given below, ABCD is a quadrilateral in which AB = AD, ∠A = 90° =∠C, BC = 8 cm and CD = 6 cm. Find AB and calculate the area of ∆ ABD.

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Q15 (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.

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Q16 (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 AB2 = 4AD² - 3AC².

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Q17 In ∆ ABC, AB = AC = x, BC = 10 cm and the area of ∆ ABC is 60 cm². Find x.

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Q18 In a rhombus, If diagonals are 30 cm and 40 cm, find its perimeter.

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Q19 (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.

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Q20 AD is perpendicular to the side BC of an equilateral Δ ABC. Prove that 4AD² = 3AB².

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Q21 In figure (i) given below, D and E are mid-points of the sides BC and CA respectively of a ΔABC, right angled at C.

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Q22 If AD, BE and CF are medians of EABC, prove that 3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).

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Q23 (a) In fig. (i) given below, the diagonals AC and BD of a quadrilateral ABCD intersect at O, at right angles. Prove that
AB² + CD² = AD² + BC².

(b) In figure (ii) given below, OD⊥BC, OE ⊥CA and OF ⊥ AB. Prove that :

(i) OA² + OB² + OC² = AF² + BD² + CE² + OD² + OE² + OF².
(ii) OAF² + BD² + CE² = FB² + DC² + EA².

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Q24 In a quadrilateral, ABCD∠B = 90° = ∠D. Prove that 2 AC² - BC2 = AB² + AD² + DC².

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Q25 In a ∆ ABC, ∠ A = 90°, CA = AB and D is a point on AB produced. Prove that : DC² - BD² = 2AB. AD.

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Q26 In an isosceles triangle ABC, AB = AC and D is a point on BC produced. Prove that AD² = AC² + BD.CD.

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Pythagoras Theorem Exercise 12

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Q1 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

Q2 Foot of a 10 m long ladder leaning against a vertical well is 6 m away from the base of the wail. Find the height of the point on the wall where the top of the ladder reaches

Q3 A guy attached a wire 24 m long to a vertical pole of height 18 m 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 taught?

Q4 Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

Q5 In a right-angled triangle, if hypotenuse is 20 cm and the ratio of the other two sides is 4:3, find the sides.

Q6 If the sides of a triangle are in the ratio 3:4:5, prove that it is right-angled triangle.

Q8 The hypotenuse of right triangle is 6m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.

Q9 ABC is an isosceles triangle right angled at C. Prove that AB² = 2AC².

Q10 In a triangle ABC, AD is perpendicular to BC. Prove that AB² + CD² = AC² + BD².

Q11 In ∆PQR, PD ⊥ QR, such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b) (a - b) = (c + d) (c - d).

Q12 ABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. Find the altitude on BC and Hence, calculate its area.

Q13 Find the area and the perimeter of a square whose diagonal is 10 cm long.

Q16 (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 AB2 = 4AD² - 3AC².

Q17 In ∆ ABC, AB = AC = x, BC = 10 cm and the area of ∆ ABC is 60 cm². Find x.

Q18 In a rhombus, If diagonals are 30 cm and 40 cm, find its perimeter.

Q20 AD is perpendicular to the side BC of an equilateral Δ ABC. Prove that 4AD² = 3AB².

Q21 In figure (i) given below, D and E are mid-points of the sides BC and CA respectively of a ΔABC, right angled at C.

Q22 If AD, BE and CF are medians of EABC, prove that 3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).

Q24 In a quadrilateral, ABCD∠B = 90° = ∠D. Prove that 2 AC² - BC2 = AB² + AD² + DC².

Q25 In a ∆ ABC, ∠ A = 90°, CA = AB and D is a point on AB produced. Prove that : DC² - BD² = 2AB. AD.

Q26 In an isosceles triangle ABC, AB = AC and D is a point on BC produced. Prove that AD² = AC² + BD.CD.

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Pythagoras Theorem Exercise 12

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Q1 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

Q2 Foot of a 10 m long ladder leaning against a vertical well is 6 m away from the base of the wail. Find the height of the point on the wall where the top of the ladder reaches

Q3 A guy attached a wire 24 m long to a vertical pole of height 18 m 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 taught?

Q4 Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.

Q5 In a right-angled triangle, if hypotenuse is 20 cm and the ratio of the other two sides is 4:3, find the sides.

Q6 If the sides of a triangle are in the ratio 3:4:5, prove that it is right-angled triangle.

Q8 The hypotenuse of right triangle is 6m more than twice the shortest side. If the third side is 2m less than the hypotenuse, find the sides of the triangle.

Q9 ABC is an isosceles triangle right angled at C. Prove that AB² = 2AC².

Q10 In a triangle ABC, AD is perpendicular to BC. Prove that AB² + CD² = AC² + BD².

Q11 In ∆PQR, PD ⊥ QR, such that D lies on QR. If PQ = a, PR = b, QD = c and DR = d, prove that (a + b) (a - b) = (c + d) (c - d).

Q12 ABC is an isosceles triangle with AB = AC = 12 cm and BC = 8 cm. Find the altitude on BC and Hence, calculate its area.

Q13 Find the area and the perimeter of a square whose diagonal is 10 cm long.

Q16 (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 AB2 = 4AD² - 3AC².

Q17 In ∆ ABC, AB = AC = x, BC = 10 cm and the area of ∆ ABC is 60 cm². Find x.

Q18 In a rhombus, If diagonals are 30 cm and 40 cm, find its perimeter.

Q20 AD is perpendicular to the side BC of an equilateral Δ ABC. Prove that 4AD² = 3AB².

Q21 In figure (i) given below, D and E are mid-points of the sides BC and CA respectively of a ΔABC, right angled at C.

Q22 If AD, BE and CF are medians of EABC, prove that 3(AB² + BC² + CA²) = 4(AD² + BE² + CF²).

Q24 In a quadrilateral, ABCD∠B = 90° = ∠D. Prove that 2 AC² - BC2 = AB² + AD² + DC².

Q25 In a ∆ ABC, ∠ A = 90°, CA = AB and D is a point on AB produced. Prove that : DC² - BD² = 2AB. AD.

Q26 In an isosceles triangle ABC, AB = AC and D is a point on BC produced. Prove that AD² = AC² + BD.CD.

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Q1 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

Q16 (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 AB2 = 4AD² - 3AC².