Pythagoras Theorem
ICSE Class 9 · Mathematics · Chapter 12
What this chapter covers
- Right-angled triangles and the words you need
- Statement of Pythagoras theorem
- Area-based proof: four triangles in a square
- Area-based proof: squares on the sides
- Finding an unknown side
- Applications in figures and everyday situations
- Proving results with Pythagoras theorem
- The converse of Pythagoras theorem
- Applications of the converse
- Common mistakes to avoid
Key points
- The hypotenuse is the side opposite the right angle and is the longest side of a right-angled triangle.
- Pythagoras theorem: in a right-angled triangle, c2 = a2 + b2, where c is the hypotenuse.
- To find a leg, subtract: a2 = c2 - b2.
- Four-triangle proof: a square of side a + b holds four right-angled triangles and a square of side c, so (a + b)2 = c2 + 2ab, which gives c2 = a2 + b2.
- Squares-on-sides proof: the perpendicular from the right-angle vertex splits the square on the hypotenuse into two rectangles equal in area to the squares on the legs, using congruent triangles (SAS) and the half-rectangle rule for triangle areas.
- Converse: if c2 = a2 + b2 for the longest side c, the angle opposite c is 90°; it is proved by SSS congruence with a constructed right-angled triangle.
- Use the theorem to find lengths from a right angle, and the converse to prove a right angle from lengths.
- A Pythagorean triplet is three whole numbers with a2 + b2 = c2; any whole-number multiple of a triplet is also a triplet.
- Common figures with hidden right angles: a wall and the ground, the diagonal of a rectangle, the altitude of an isosceles or equilateral triangle, the diagonals of a rhombus.
- Never write c = a + b; never compare any square but the longest side's with the sum of the other two.
Questions answered
State Pythagoras theorem.
In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
State the converse of Pythagoras theorem.
If the square on one side of a triangle is equal to the sum of the squares on the other two sides, then the angle opposite the first side is a right angle.
In △ XYZ, ∠ X = 90°. Name the hypotenuse and write the relation given by Pythagoras theorem.
The hypotenuse is YZ, the side opposite the right angle, and YZ2 = XY2 + XZ2.
Why is the hypotenuse always the longest side of a right-angled triangle?
The right angle is the greatest angle of the triangle, because the other two angles add up to only 90°. In a triangle the greater angle has the greater side opposite it, so the side opposite the right angle is the longest.
The legs of a right-angled triangle are 20 cm and 21 cm. Find the hypotenuse.
202 + 212 = 400 + 441 = 841, so the hypotenuse is √(841) = 29 cm.
The hypotenuse of a right-angled triangle is 17 cm and one leg is 8 cm. Find the other leg.
172 - 82 = 289 - 64 = 225, so the other leg is 15 cm.
Find the diagonal of a square of side 7 cm, in surd form.
Diagonal2 = 72 + 72 = 98, so the diagonal is √(98) = 7√(2) cm.
A kite string is 50 m long and is tied to a peg on the ground. The kite is directly above a point 30 m from the peg. Assuming the string is straight, how high is the kite?
Height2 = 502 - 302 = 2500 - 900 = 1600, so the kite is 40 m high.
Past board exam questions
ICSE 2019 sample paper · 4 marks
Triangle ABC is a right angle triangle with ∠B = 90° and D is mid point of side BC. AD is joined. Prove that: AC² = AD² + 3CD²
Show answer
BD = CD (D is the mid-point of BC), so BC = 2CD.
In right △ABC: AC² = AB² + BC² = AB² + 4CD².
In right △ABD: AD² = AB² + BD² = AB² + CD², so AB² = AD² − CD².
∴ AC² = AD² − CD² + 4CD² = AD² + 3CD². Hence proved.