Parts of a right triangle
A right triangle has one angle of exactly 90°. The side across from the right angle is the hypotenuse. It is always the longest side. The other two are the legs.
The three angles of any triangle add up to 180°. One is 90°, so the other two (acute angles) add up to 90°. If one is 35°, the other is 55°.
The relation between the sides is the Pythagoras theorem: a² + b² = c². See Pythagoras theorem.
Median to the hypotenuse
A median joins a corner to the middle of the opposite side. In a right triangle with right angle at O, let M be the middle of the hypotenuse.
Rule: OM = ½ × hypotenuse.
Why? Put a second copy of the triangle against the first to make a rectangle. In a rectangle, the two diagonals are equal and cut each other in half. One diagonal is the hypotenuse. The other is double of OM. So OM = half the hypotenuse. This also means M is the same distance from all three corners, so a circle with centre M passes through all three.
Backward rule: if a median of a triangle is half the side it goes to, the angle at the opposite corner is 90°.
The 30° rule
In a right triangle with a 30° angle, the leg opposite the 30° angle is half the hypotenuse.
Why? Flip the triangle over the other leg. The two joined triangles make a bigger triangle with all angles 60°, so it is equilateral, and all its sides are equal to the hypotenuse. The small leg is half of one side of this equilateral triangle.
Backward rule: if a leg is half the hypotenuse, the angle opposite it is 30°. The other acute angle is 60°.
Congruence of right triangles
Two triangles are congruent if one fits exactly on the other. For right triangles the right angle is already equal (90° each), so we need just two more matching facts. Four tests:
- HL: hypotenuse and one leg are equal.
- LL: both legs are equal.
- LA: one leg and one acute angle are equal.
- HA: hypotenuse and one acute angle are equal.
Careful: with LA the leg can be next to the angle or across from it; both work for right triangles.
Try it: fold and cut
Cut a rectangle from paper. Draw one diagonal and cut along it: you get two right triangles that fit exactly (congruent). Now draw the other diagonal on the whole rectangle. Predict: how do the two diagonals compare? Measure with a ruler. They are equal and cross at their middle. That is the median rule.
Key formulas and definitions
- Acute angles: A + B = 90°
- Median to hypotenuse: OM = c / 2
- Leg opposite 30° = c / 2
- a² + b² = c²
- Congruence: HL, LL, LA, HA
Worked examples
1. The hypotenuse of a right triangle is 14 cm. Find the median to it.
Median = half the hypotenuse = 14 / 2 = 7 cm.
2. The median to the hypotenuse is 6.5 cm. Find the hypotenuse.
Hypotenuse = 2 x 6.5 = 13 cm.
3. One acute angle is 35°. Find the other.
The two acute angles add up to 90°. Other = 90° - 35° = 55°.
4. A right triangle has a 30° angle and hypotenuse 18 cm. Find the leg opposite 30°.
Opposite leg = half the hypotenuse = 9 cm.
5. The leg opposite 30° is 8 cm. Find the hypotenuse and then the other leg.
Hypotenuse = 2 x 8 = 16 cm. Other leg: 16² - 8² = 256 - 64 = 192, so leg = √192 = 8√3 ≈ 13.86 cm.
6. The median to the hypotenuse is 5 cm and one leg is 6 cm. Find the other leg.
Hypotenuse = 2 x 5 = 10 cm. Other leg² = 10² - 6² = 100 - 36 = 64, so other leg = 8 cm.
7. Triangle 1 has hypotenuse 13 and a leg 5. Triangle 2 has hypotenuse 13 and a leg 5. Are they congruent? Find the third side.
Yes, by HL (right angle, equal hypotenuse and equal leg). Third side² = 13² - 5² = 169 - 25 = 144, so third side = 12.
Common mistakes
- Saying the median is half of any side. It is half only for the hypotenuse of a right triangle.
- Putting the 30° half-rule on the wrong leg. It is the leg across from (opposite) 30°, not next to it.
- Using "SSA" for every triangle. HL works only because the angle is 90°.
- Calling a side the hypotenuse when it is not across from the right angle.