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The Triangle and its Properties - Right-angled Triangles and Pythagoras Property

Grade 7CBSE

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 of the angles is exactly 90∘90^{\circ}. Visually, this is indicated by a small square symbol at the vertex where the two perpendicular sides meet, forming a perfect 'L' shape.

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The side opposite the right angle is called the hypotenuse. It is the longest side of a right-angled triangle. The other two sides that form the right angle are known as the legs, often referred to as the 'base' and the 'altitude' or 'height'.

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The Pythagoras Property states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If you imagine a square drawn 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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The Pythagoras Property only holds true for right-angled triangles. This means if the property a2+b2=c2a^2 + b^2 = c^2 is satisfied by the side lengths of a triangle, the triangle must contain a right angle (90∘90^{\circ}) opposite the side cc.

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Pythagorean Triplets are sets of three positive integers (a,b,c)(a, b, c) that satisfy the rule a2+b2=c2a^2 + b^2 = c^2. Common examples include (3,4,5)(3, 4, 5) and (5,12,13)(5, 12, 13). These triplets are useful for quickly identifying right-angled triangles in geometry problems.

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In any right-angled triangle, if you know the lengths of any two sides, you can calculate the third side using the Pythagoras property. For example, if the triangle is oriented with the hypotenuse slanted, you can still find its length by squaring the horizontal base and the vertical height.

📐Formulae

Pythagoras Theorem: a2+b2=c2a^2 + b^2 = c^2

Calculating the Hypotenuse: c=a2+b2c = \sqrt{a^2 + b^2}

Calculating a Leg: a=c2−b2a = \sqrt{c^2 - b^2} or b=c2−a2b = \sqrt{c^2 - a^2}

Condition for Right-angled Triangle: Side12+Side22=Longest Side2Side_1^2 + Side_2^2 = Longest\,Side^2

💡Examples

Problem 1:

Find the length of the hypotenuse of a right-angled triangle whose legs are 6 cm6\text{ cm} and 8 cm8\text{ cm} long.

Solution:

  1. Let the legs be a=6 cma = 6\text{ cm} and b=8 cmb = 8\text{ cm}. Let the hypotenuse be cc.
  2. According to Pythagoras Property: a2+b2=c2a^2 + b^2 = c^2
  3. Substitute the values: 62+82=c26^2 + 8^2 = c^2
  4. Calculate the squares: 36+64=c236 + 64 = c^2
  5. Add the values: 100=c2100 = c^2
  6. Find the square root: c=100=10 cmc = \sqrt{100} = 10\text{ cm}.

Explanation:

We use the standard Pythagoras formula where the sum of the squares of the two shorter sides gives the square of the longest side (hypotenuse).

Problem 2:

A 13 m13\text{ m} long ladder is placed against a wall such that its foot is 5 m5\text{ m} away from the wall. At what height does the ladder reach the wall?

Solution:

  1. Here, the ladder acts as the hypotenuse (c=13 mc = 13\text{ m}) and the distance from the wall is the base (a=5 ma = 5\text{ m}).
  2. We need to find the height (bb).
  3. Using a2+b2=c2a^2 + b^2 = c^2, we get: 52+b2=1325^2 + b^2 = 13^2
  4. 25+b2=16925 + b^2 = 169
  5. Subtract 2525 from both sides: b2=169−25b^2 = 169 - 25
  6. b2=144b^2 = 144
  7. b=144=12 mb = \sqrt{144} = 12\text{ m}.

Explanation:

This is a real-world application of the Pythagoras property where the wall, the ground, and the ladder form a right-angled triangle. We solve for the missing leg (the height on the wall).