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Geometry - Pythagoras' Theorem and its applications

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Right-Angled Triangle is a triangle in which one interior angle is exactly 90∘90^{\circ}. Visually, this is represented by two sides meeting to form a perfect 'L' shape, with a small square symbol drawn at the vertex to indicate the right angle.

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The Hypotenuse is the longest side of a right-angled triangle and is always the side directly opposite the right angle. In a drawing, if the right angle is at the bottom corner, the hypotenuse is the slanting diagonal line connecting the top of the vertical side to the end of the horizontal side.

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The two sides that form the right angle are known as the 'Base' and the 'Perpendicular' (or height). Together, they are sometimes called the 'legs' of the triangle.

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Pythagoras' Theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This can be visualized by drawing squares on each side of the triangle; the area of the largest square on the hypotenuse equals the combined area of the two smaller squares.

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A Pythagorean Triplet is a set of three positive integers (a,b,c)(a, b, c) that satisfy the formula a2+b2=c2a^{2} + b^{2} = c^{2}. For example, (3,4,5)(3, 4, 5) and (5,12,13)(5, 12, 13) are triplets because they represent the exact whole-number side lengths of a right-angled triangle.

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The Converse of Pythagoras' Theorem is a way to test if a triangle is right-angled. If the side lengths a,b,a, b, and cc satisfy a2+b2=c2a^{2} + b^{2} = c^{2}, then the angle opposite side cc must be a right angle.

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Practical applications often involve visualizing real-world objects as parts of a triangle. For example, a vertical wall and the horizontal ground form the two legs of a right angle, while a ladder leaning against the wall serves as the hypotenuse.

📐Formulae

c2=a2+b2c^{2} = a^{2} + b^{2}

Hypotenuse2=Base2+Perpendicular2Hypotenuse^{2} = Base^{2} + Perpendicular^{2}

c=a2+b2c = \sqrt{a^{2} + b^{2}}

a=c2−b2a = \sqrt{c^{2} - b^{2}}

b=c2−a2b = \sqrt{c^{2} - a^{2}}

💡Examples

Problem 1:

A ladder is placed against a wall such that its foot is 55 m away from the wall. If the ladder reaches a window 1212 m high on the wall, find the length of the ladder.

Solution:

  1. Identify the given values: Base (aa) = 55 m, Perpendicular (bb) = 1212 m. We need to find the length of the ladder, which is the Hypotenuse (cc).
  2. Use the formula: c2=a2+b2c^{2} = a^{2} + b^{2}
  3. Substitute the values: c2=52+122c^{2} = 5^{2} + 12^{2}
  4. Calculate the squares: c2=25+144c^{2} = 25 + 144
  5. Add the results: c2=169c^{2} = 169
  6. Find the square root: c=169=13c = \sqrt{169} = 13 m.

Explanation:

The wall and ground form a right angle. The ladder forms the hypotenuse. By squaring the distances from the wall and the height of the window, we find the square of the ladder's length.

Problem 2:

The hypotenuse of a right-angled triangle is 1010 cm and one of its legs is 88 cm. Find the length of the third side.

Solution:

  1. Identify the given values: Hypotenuse (cc) = 1010 cm, Leg (aa) = 88 cm. We need to find the other Leg (bb).
  2. Use the modified formula: b2=c2−a2b^{2} = c^{2} - a^{2}
  3. Substitute the values: b2=102−82b^{2} = 10^{2} - 8^{2}
  4. Calculate the squares: b2=100−64b^{2} = 100 - 64
  5. Subtract the values: b2=36b^{2} = 36
  6. Find the square root: b=36=6b = \sqrt{36} = 6 cm.

Explanation:

When we know the hypotenuse and one side, we subtract the square of the known side from the square of the hypotenuse to solve for the missing side.