Right-Angled Triangle Property
A right-angled triangle (or right triangle) has one angle equal to 90°. The side opposite the right angle is the longest side and is called the hypotenuse.
The other two sides are called the legs (or perpendicular and base). Since one angle is already 90°, the other two angles must add up to 90° (they are complementary).
Right triangles are fundamental in geometry. They appear in construction, navigation, and form the basis of trigonometry.
What is Right-Angled Triangle Property - Grade 7 Maths (Triangles)?
Definition: A right-angled triangle is a triangle in which one angle is exactly 90°.
Key terms:
- Hypotenuse: The side opposite the right angle — always the longest side.
- Legs: The two sides that form the right angle.
Properties:
- One angle = 90°, the other two angles are complementary (sum to 90°).
- The hypotenuse is always longer than each leg.
- The Pythagoras Theorem applies: hypotenuse² = leg₁² + leg₂².
Right-Angled Triangle Property Formula
Pythagoras Theorem:
c² = a² + b²
Where c = hypotenuse, a and b = legs.
Other Properties:
- Sum of acute angles: ∠A + ∠B = 90° (where ∠C = 90°)
- Area = (1/2) × leg₁ × leg₂
- The median to the hypotenuse = half the hypotenuse
Types and Properties
Types of Right Triangles:
- Isosceles right triangle: legs are equal, angles are 45°-45°-90°.
- Scalene right triangle: all sides different, e.g., 30°-60°-90° or 3-4-5 triangle.
Common Pythagorean Triples:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Solved Examples
Example 1: Finding the Hypotenuse
Problem: A right triangle has legs of 3 cm and 4 cm. Find the hypotenuse.
Solution:
- c² = 3² + 4² = 9 + 16 = 25
- c = √25 = 5 cm
Answer: Hypotenuse = 5 cm.
Example 2: Finding a Leg
Problem: Hypotenuse = 13 cm, one leg = 5 cm. Find the other leg.
Solution:
- 13² = 5² + b²
- 169 = 25 + b²
- b² = 144
- b = 12 cm
Answer: The other leg = 12 cm.
Example 3: Finding Acute Angles
Problem: One acute angle of a right triangle is 35°. Find the other.
Solution:
- Sum of acute angles = 90°
- Other angle = 90° − 35° = 55°
Answer: The other acute angle = 55°.
Example 4: Area Calculation
Problem: A right triangle has legs 6 cm and 8 cm. Find its area.
Solution:
- Area = (1/2) × 6 × 8 = 24 cm²
Answer: Area = 24 cm².
Example 5: Checking for Right Angle
Problem: A triangle has sides 6, 8, and 10. Is it right-angled?
Solution:
- Check: 10² = 100, 6² + 8² = 36 + 64 = 100
- Since 10² = 6² + 8², it satisfies Pythagoras Theorem.
Answer: Yes, it is a right-angled triangle..
Example 6: Word Problem — Ladder
Problem: A ladder 10 m long leans against a wall. Its foot is 6 m from the wall. How high up the wall does it reach?
Solution:
- Hypotenuse (ladder) = 10 m, base = 6 m
- 10² = 6² + h²
- 100 = 36 + h²
- h² = 64, h = 8 m
Answer: The ladder reaches 8 m up the wall.
Real-World Applications
Real-world uses:
- Construction: Builders use the 3-4-5 rule to make sure corners are exactly 90°.
- Navigation: Finding shortest distances using right triangle calculations.
- Ladders: Finding how high a ladder reaches against a wall.
- Ramps: Calculating the length of a ramp given its height and horizontal distance.
Key Points to Remember
- One angle is 90°; the other two are complementary.
- The hypotenuse is the longest side (opposite the right angle).
- Pythagoras Theorem: c² = a² + b².
- Area = (1/2) × leg₁ × leg₂.
- If c² = a² + b², the triangle is right-angled (converse of Pythagoras).
- The median to the hypotenuse equals half the hypotenuse.
- Common triples: 3-4-5, 5-12-13, 8-15-17.
Practice Problems
- Find the hypotenuse if legs are 8 cm and 15 cm.
- Hypotenuse = 25 cm, one leg = 7 cm. Find the other leg.
- One acute angle is 42°. Find the other.
- Is a triangle with sides 9, 12, 15 right-angled?
- A 13 m ladder has its foot 5 m from a wall. Find the height reached.
Frequently Asked Questions
Q1. What is a right-angled triangle?
A triangle with one angle exactly 90°. The side opposite the right angle is the hypotenuse.
Q2. What is the hypotenuse?
The longest side of a right triangle, opposite the right angle.
Q3. What is the Pythagoras Theorem?
In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: c² = a² + b².
Q4. How do you check if a triangle is right-angled?
Check if the square of the longest side equals the sum of squares of the other two sides. If yes, it is right-angled.
Q5. Can a right triangle be isosceles?
Yes. If both legs are equal, the triangle is a 45°-45°-90° isosceles right triangle.
Related Topics
- Pythagoras Theorem
- Classification of Triangles
- Angle Sum Property of Triangle
- Congruence of Triangles
- Exterior Angle Property of Triangle
- Properties of Isosceles Triangle
- Properties of Equilateral Triangle
- Triangle Inequality Property
- Medians and Altitudes of Triangle
- Congruent Triangles - Proofs
- Inequalities in Triangles
- Similar Triangles
- Basic Proportionality Theorem (BPT)
- Converse of Basic Proportionality Theorem
