**Chapter 13 - Pythagoras' Theorem**

**Practice set 48**

**Practice set 48**

**1. In the figures below, find the value of ‘x’.**

**(i) **

**Given:**

**LM = 7 units**

**MN = 24 units**

**∠LMN = 90°**

** **

**To find:**

**The value of x**

** **

**Solution:**

**In ∆LMN, ∠LMN = 90°**

**∴ Side LN is the hypotenuse**

** **

**According to Pythagoras’ theorem,**

**l(LN)² = l(LM)² + l(MN)²**

**∴ x² = 7² + 24²**

**∴ x² = 49 + 576**

**∴ x² = 625**

**∴ x² = 25²**

**∴ x = 25 units**

** **

**Ans: **The value of x is 25 units.

**(ii)**

**Given:**

**PQ = 9 units**

**PR = 41 units**

**∠PQR = 90°**

** **

**To find:**

**The value of x**

** **

**Solution:**

**In ∆PQR, ∠PQR = 90°**

**∴ Side PR is the hypotenuse**

** **

**According to Pythagoras’ theorem,**

**l(PR)² = l(PQ)² + l(QR)²**

**∴ 41² = 9² + x²**

**∴ 1681 = 81 + x²**

**∴ 1681 – 81 = x²**

**∴ x² = 1600**

**∴ x² = 40²**

**∴ x = 40 units**

** **

**Ans: **The value of x is 40 units.

**(iii)**

**Given:**

**DF = 8 units**

**EF = 17 units**

**∠EDF = 90°**

** **

**To find:**

**The value of x**

** **

**Solution:**

**In ∆EDF, ∠EDF = 90°**

**∴ Side EF is the hypotenuse**

** **

**According to Pythagoras’ theorem,**

**l(EF)² = l(ED)² + l(DF)²**

**∴ 17² = x² + 8²**

**∴ 289 = x² + 64**

**∴ 289 – 64 = x²**

**∴ x² = 225**

**∴ x² = 15²**

**∴ x = 15 units**

** **

**Ans: **The value of x is 15 units.

**2. In the right-angled ∆PQR, ∠P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.**

**Given:**

**∠P = 90°**

**l(PQ) = 24 cm**

**l(PR) = 10 cm**

** **

**To find:**

**l(QR)**

** **

**Solution:**

**In ∆PQR, ∠P = 90°**

**∴ Side QR is the hypotenuse**

** **

**According to Pythagoras’ theorem,**

**l(QR)² = l(PR)² + l(PQ)²**

**∴ l(QR)² = 10² + 24²**

**∴ l(QR)² = 100 + 576**

**∴ l(QR)² = 676**

**∴ l(QR)² = 26²**

**∴ l(QR) = 26 cm**

**Ans:** The length of side QR is 26 cm. ** **

**3. In the right-angled ∆LMN, ∠M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.**

**Given:**

**∠M = 90°**

**l(LM) = 12 cm**

**l(LN) = 20 cm**

** **

**To find:**

**l(MN) **

** **

**Solution:**

**In ∆LMN, ∠M = 90°**

**∴ Side LN is the hypotenuse**

** **

**According to Pythagoras’ theorem,**

**l(LN)² = l(LM)² + l(MN)²**

**∴ 20² = 12² + l(MN)²**

**∴ l(MN)² = 20² – 12²**

**∴ l(MN)² = 400 – 144**

**∴ l(MN)² = 256**

**∴ l(MN)² = 16²**

**∴ l(MN)= 16 cm**

** **

**Ans: **The length of seg MN is 16 cm.

**4. The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder ?**

**Given:**

**∠ABC = 90°**

**The top of the ladder i.e. l(AC) = 15 m**

**It reaches a window i.e. l(AB) = 9 m**

** **

**To find:**

**The distance between the base of the wall and that of the ladder i.e. l(BC)**

** **

**Solution:**

**The wall and the ground are perpendicular to each other and the ladder reaching the wall forms the hypotenuse to the right-angled triangle.**

** **

**In ∆ABC, ∠ABC = 90°**

**∴ Side AC is the hypotenuse**

** **

**According to Pythagoras’ theorem,**

**l(AC)² = l(AB)² + l(BC)²**

**∴ 15² = l(BC)² + 9²**

**∴ 225 = l(BC)² + 81**

**∴ 225 – 81 = l(BC)²**

**∴ l(BC)² = 144**

**∴ l(BC)² = 12²**

**∴ l(BC) = 12 m**

** **

**Ans: **The distance between the base of the wall and that of the ladder is 12 m.

**Practice set 49**

**Practice set 49****1. Find the Pythagorean triplets from among the following sets of numbers. **

**(i) 3, 4, 5 **

**Given:**

**The numbers given are 3, 4, 5**

** **

**To find:**

**Whether the numbers are a Pythagorean triplet**

** **

**Solution:**

**The biggest of the three numbers is 5**

**Let 5 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**5² = 3² + 4²**

**∴ 25 = 9 + 16**

**∴ 25 = 25 which is true**

**Ans:** 3, 4, 5 is a Pythagorean triplet.

**(ii) 2, 4, 5**

**Given:**

**The numbers given are 2, 4, 5**

** **

**To find:**

**Whether the numbers is a Pythagorean triplet**

** **

**Solution:**

**The biggest of the three numbers is 5**

**Let 5 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**5² = 2² + 4²**

**∴ 25 = 4 + 16**

**∴ 25 ≠ 20 **

** **

**Ans:** 2, 4, 5 is not a Pythagorean triplet.

**(iii) 4, 5, 6**

**Given:**

**The numbers given are 4, 5, 6**

** **

**To find:**

**Whether the numbers are a Pythagorean triplet**

** **

**Solution:**

**The biggest of the three numbers is 6**

**Let 6 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**6² = 4² + 5²**

**∴ 36 = 16 + 25**

**∴ 36 ≠ 41 **

**Ans:** 4, 5, 6 is not a Pythagorean triplet.

**(iv) 2, 6, 7**

**Given:**

**The numbers given are 2, 6, 7**

** **

**To find:**

**Whether the numbers are a Pythagorean triplet**

** **

**Solution:**

**The biggest of the three numbers is 7**

**Let 7 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**7² = 2² + 6²**

**∴ 49 = 4 + 36**

**∴ 49 ≠ 40 **

** **

**Ans:** 2, 6, 7 is not a Pythagorean triplet.

**(v) 9, 40, 41**

**Given:**

**The numbers given are 9, 40, 41**

** **

**To find:**

**Whether the numbers are a Pythagorean triplet**

** **

**Solution:**

**The biggest of the three numbers is 41**

**Let 41 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**41² = 9² + 40²**

**∴ 1681 = 81 + 1600**

**∴ 1681 = 1681 which is true**

** **

**Ans:** 9, 40, 41 is a Pythagorean triplet.

**(vi) 4, 7, 8**

**Given:**

**The numbers given are 4, 7, 8**

** **

**To find:**

**Whether the numbers are a Pythagorean triplet**

** **

**Solution:**

**The biggest of the three numbers is 8**

**Let 8 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**8² = 4² + 7²**

**∴ 64 = 16 + 49**

**∴ 64 ≠ 65 **

** **

**Ans:** 4, 7, 8 is not a Pythagorean triplet.

**2. The sides of some triangles are given below. Find out which ones are right-angled triangles? **

**(i) 8, 15, 17 **

**Given:**

**The sides of a triangle are 8, 15, 17**

** **

**To find:**

**Whether the sides are of a right-angled triangle**

** **

**Solution:**

**The biggest of the three numbers is 17**

**Let 17 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**17² = 8² + 15²**

**∴ 289 = 64 + 225**

**∴ 289 = 289 **

** **

**Since the given numbers satisfies the pythagorean theorem**

**∴ The sides will form a right-angled triangle**

** **

**Ans: **The sides of lengths 8,15,17 will form a right-angled triangle.

**(ii) 11, 12, 15 **

**Given:**

**The sides of a triangle are 11, 12, 15**

** **

**To find:**

**Whether the sides are of a right-angled triangle**

** **

**Solution:**

**The biggest of the three numbers is 15**

**Let 15 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**15² = 11² + 12²**

**∴ 225 = 121 + 144**

**∴ 225 ≠ 265 **

** **

**Since the given numbers does not satisfy the pythagorean theorem**

**∴ The sides will not form a right-angled triangle**

** **

**Ans: **The sides of lengths 11, 12, 15 will not form a right-angled triangle.

**(iii) 11, 60, 61 **

**Given:**

**The sides of a triangle are 11, 60, 61**

** **

**To find:**

**Whether the sides are of a right-angled triangle**

** **

**Solution:**

**The biggest of the three numbers is 61**

**Let 61 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**61² = 11² + 60²**

**∴ 3721 = 121 + 3600**

**∴ 3721 = 3721 **

** **

**Since the given numbers satisfies the pythagorean theorem**

**∴ The sides will form a right-angled triangle**

** **

**Ans: **The sides of lengths 11, 60, 61 will form a right-angled triangle.

**(iv) 1.5, 1.6, 1.7**

**Given:**

**The sides of a triangle are 1.5, 1.6, 1.7**

** **

**To find:**

**Whether the sides are of a right-angled triangle**

** **

**Solution:**

**The biggest of the three numbers is 1.7**

**Let 1.7 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**1.7² = 1.5² + 1.6²**

**∴ 2.89 = 2.25 + 2.56**

**∴ 2.89 ≠ 4.81 **

** **

**Since the given numbers does not satisfy the pythagorean theorem**

**∴ The sides will not form a right-angled triangle**

** **

**Ans: **The sides of lengths 1.5, 1.6, 1.7 will not form a right-angled triangle.

**(v) 40, 20, 30**

**Given:**

**The sides of a triangle are 40, 20, 30**

** **

**To find:**

**Whether the sides are of a right-angled triangle**

** **

**Solution:**

**The biggest of the three numbers is 40**

**Let 40 be the hypotenuse.**

** **

**According to Pythagoras’ theorem,**

**40² = 20² + 30²**

**∴ 1600 = 400 + 900**

**∴ 1600 ≠ 1300**

** **

**Since the given numbers does not satisfy the pythagorean theorem**

**∴ The sides will not form a right-angled triangle**

** **

**Ans: **The sides of lengths 40, 20, 30 will not form a right-angled triangle.