Pythagoras Theorem — Class 10 Maharashtra SSC Mathematics revision notes
A 4-page visual revision sheet for Pythagoras Theorem, written for Class 10 Maharashtra SSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Pythagoras Theorem in one look
the right angle fixes the relation between the three sides, and the converse tests for it
The theorem and its converse
- theorem: in a right angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides
- the hypotenuse is the side opposite the right angle, and it is always the longest of the three
- converse: if in a triangle the square of one side equals the sum of the squares of the other two, then the angle opposite that first side is a right angle
- the theorem starts from a right angle and gives a relation. The converse starts from the relation and proves the right angle
- which one a question needs is decided by what is given: an angle given means the theorem, three lengths given means the converse
The relations worth knowing by heart
- the theorem itself, c being the hypotenuse
a^2 + b^2 = c^2 - finding a shorter side instead
a = √c^2 - b^2 - the altitude on the hypotenuse, from similarity
p^2 = xy - the diagonal of a rectangle of sides l and b
d = √l^2 + b^2 - the altitude of an equilateral triangle of side a
h = frac√32a - the diagonal of a square of side a
d = a√2
Pythagorean triplets, and why they save time
- three whole numbers satisfying a^2 + b^2 = c^2 form a Pythagorean triplet
- the ones worth recognising on sight: 3, 4, 5 and 5, 12, 13 and 8, 15, 17 and 7, 24, 25
- any multiple of a triplet is also a triplet, so 6, 8, 10 and 9, 12, 15 and 15, 20, 25 all come from 3, 4, 5
- spotting a triplet means the answer is available without a square root, which is where the time is saved
- the largest number is always the hypotenuse, so in 5, 12, 13 the right angle sits opposite 13
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Pythagoras Theorem sheet covers
- Pythagoras theorem
- Converse of Pythagoras theorem
- The theorem and its converse
- The relations worth knowing by heart
- the theorem itself, c being the hypotenuse
- finding a shorter side instead
- the altitude on the hypotenuse, from similarity
- the diagonal of a rectangle of sides l and b
- the altitude of an equilateral triangle of side a
- the diagonal of a square of side a
- Pythagorean triplet
- Pythagorean triplets, and why they save time
- Proof of Pythagoras theorem
- Similarity in a right triangle
- Proving the theorem through similarity
- The altitude on the hypotenuse
- Applying the converse
- A height and distance problem
- Where this theorem turns up elsewhere
- Classifying a triangle from its three sides
- Terms in this chapter
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Pythagoras Theorem the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.