Last reviewed: 11 November 2026
Updated November 2026. These Grade 7 Pythagorean relationship notes show how to name the sides of a right-angled triangle, how to write the relationship a2 + b2 = c2, and how to use it when a ladder leans on a wall or a path cuts across a rectangle. In a right-angled triangle the side opposite the right angle is the hypotenuse, and it is the longest of the three sides. If the shorter sides are 3 cm and 4 cm, the hypotenuse is 5 cm, because 9 + 16 = 25.
The examples are original practice for Kenyan junior school learners, not an official paper. They follow the Grade 7 Mathematics measurements sub-strand on the Pythagorean relationship in the KICD Grade Seven curriculum designs. Trigonometry, which compares sides using sine, cosine and tangent, comes later in the Grade 9 trigonometry notes. For Grade 7 topics overall, use the Grade 7 mathematics revision guide and the Grade 7 notes hub.
On this page
Sides of a Right-Angled Triangle
A right-angled triangle has one angle of 90°. The two sides that meet at that angle are often called the base and the height (or the two shorter sides). The side that faces the right angle, and joins the other two vertices, is the hypotenuse. If you count squares on a 3-by-4 grid, the square on the hypotenuse has the same area as the two smaller squares put together. That is the Pythagorean relationship: a2 + b2 = c2, where c is the hypotenuse. OpenStax Prealgebra’s section on the Pythagorean theorem states the same equation.
Useful whole-number triples to recognise are 3-4-5, 5-12-13, 6-8-10 (a multiple of 3-4-5) and 8-15-17. If a triangle has sides in one of these ratios and you already know it is right-angled, the numbers will fit the relationship.
Applying the Relationship
To find the hypotenuse, add the squares of the two shorter sides and take the square root. To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root. A ladder standing on level ground and resting against a wall is the picture used in the design: the wall and the ground meet at a right angle, the ladder is the hypotenuse.
Worked example: finding the hypotenuse
A rectangular garden measures 8 m by 15 m. A path runs diagonally from one corner to the opposite corner. The path is the hypotenuse of a right-angled triangle whose shorter sides are 8 m and 15 m. c2 = 82 + 152 = 64 + 225 = 289. c = √289 = 17 m.
Worked example: a ladder on a wall
A 13 m ladder rests on level ground with its foot 5 m from the base of a wall. How far up the wall does it reach? The ladder is the hypotenuse, so h2 + 52 = 132. h2 + 25 = 169. h2 = 144. h = √144 = 12 m.
The numbers 5, 12 and 13 are another of the triples above, which is a quick check after you have squared and subtracted.
Worked example: recognising the sides
A triangle has sides 9 cm, 12 cm and 15 cm. Is it right-angled? Test whether the two smaller squares add to the largest square: 92 + 122 = 81 + 144 = 225 = 152. Yes. The sides are a multiple of 3-4-5 (each length is multiplied by 3), and 15 cm is the hypotenuse.
Original Practice Questions With Worked Answers
- Name the hypotenuse in a right-angled triangle whose sides are 6 cm, 8 cm and 10 cm. The 10 cm side, the longest side, opposite the right angle.
- The shorter sides of a right-angled triangle are 5 cm and 12 cm. Find the hypotenuse. √(25 + 144) = √169 = 13 cm.
- A 10 m ladder reaches 8 m up a wall. How far is the foot of the ladder from the wall? √(100 − 64) = √36 = 6 m.
- A square has side 7 cm. How long is a diagonal? Two sides of 7 cm give d2 = 49 + 49 = 98, so d = √98 = 7√2 cm.
- Do the sides 7 cm, 24 cm and 26 cm form a right-angled triangle? 72 + 242 = 49 + 576 = 625 = 252, not 262 (676). They do not.
- A learner uses 132 + 52 = h2 for the ladder question above. What went wrong? 13 m is already the hypotenuse. The unknown shorter side should be found from 132 − 52, not by adding.
Cover the answers and try each question. Finding square roots of perfect squares is practised in the Grade 7 squares and square roots notes. For timed practice use the Grade 7 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
What is the Pythagorean relationship?
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a2 + b2 = c2. For sides 3, 4 and 5 that is 9 + 16 = 25.
Which side is the hypotenuse?
It is the side opposite the right angle, and it is the longest side. In a ladder-against-a-wall question the ladder itself is the hypotenuse.
How do you find a shorter side?
Subtract the square of the known shorter side from the square of the hypotenuse, then take the square root. If the hypotenuse is 13 m and one side is 5 m, the other side is √(169 − 25) = 12 m.
What is a 3-4-5 triangle?
It is a right-angled triangle whose sides are in the ratio 3 : 4 : 5, or any multiple of those numbers, such as 6-8-10 or 9-12-15. The sides satisfy the Pythagorean relationship automatically.