# Sides and angles in a triangle, and the 30-60-90 triangle · Grade 9

> Explanation, worked examples and free practice following the Israeli curriculum.

In Sides and angles in a triangle, and the 30-60-90 triangle (Grade 9) you practise 2 skills: The longest side is opposite the largest angle · The 30-60-90 triangle.

In any triangle, the larger angle faces the longer side, and the other way round: the longer side faces the larger angle. So the longest side is opposite the largest angle, and the shortest side is opposite the smallest. When two angles are given, work out the third first: the angles of a triangle add up to 180°.

A right triangle with a 30° angle is half of an equilateral triangle: the height of the equilateral triangle splits it into two such triangles. So the side opposite the 30° angle is half the hypotenuse; this is the exact value sin 30° = ½. The converse is true too: if one side of a right triangle is half the hypotenuse, the angle opposite it is 30°.

Find the third side with Pythagoras' theorem: subtract the square of the short side from the square of the hypotenuse. The result is not a whole square, so give the square root to 1 decimal place.

**Tip:** To find the longest side, look for the largest angle and the side opposite it, not the sides that meet at it.

## What you learn

### The longest side is opposite the largest angle

- Example: In triangle ABC , ∢B = 30°, ∢C = 100° . Which side is the longest?
- Options: AC · AB · BC
- Answer: AB
- Solution: The angles of a triangle add up to 180°, so: ∢A = 180° − 30° − 100° = 50° The largest angle is ∢C , so the longest side is the one facing it: AB

### The 30-60-90 triangle

- Example: (A right triangle with an angle of 30°) In the right triangle, ∢B = 30° and the hypotenuse is 8 cm. How long is side AC ?
- Answer: 4 cm
- Solution: AC is opposite the 30° angle, so: AC = 8 ÷ 2 = 4

## Not on the site yet

- Proving that the median to the hypotenuse is half of it
- The midpoint theorem in a triangle

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education's grade 9 maths curriculum (2026–27).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 9 topics](https://kixplo.com/il/en/math/grade-9/)