Constructing Triangles (RHS)

Class 7Practical Geometry

When a right angle, the hypotenuse, and one side are given, a unique right triangle can be constructed. This is the RHS construction.

What is Constructing Triangles (RHS) - Grade 7 Maths (Practical Geometry)?

Steps to construct △ABC (∠B = 90°, hypotenuse AC, side BC given):

Draw BC of the given length.
At B, construct a 90° angle.
From C, draw an arc of radius = hypotenuse (AC).
This arc cuts the perpendicular ray at A.
Join AC.

Constructing Triangles (RHS) Formula

Right angle + Hypotenuse + One side → Unique right triangle

The hypotenuse must be longer than the given side (otherwise the arc won't intersect the perpendicular).

Types and Properties

Check: Hypotenuse > given side. If not, construction is impossible.

Solved Examples

Example 1: Standard RHS Construction

Problem: Construct △ABC: ∠B = 90°, AC = 5 cm, BC = 3 cm.

Solution:

Draw BC = 3 cm.
At B, construct 90° perpendicular.
From C, draw arc of radius 5 cm.
Arc cuts the perpendicular at A.
Join AC.
Verify: AB = √(5² − 3²) = √(25 − 9) = √16 = 4 cm.

Answer: Right triangle with sides 3, 4, 5 cm.

Example 2: Isosceles Right Triangle

Problem: Construct a right triangle with hypotenuse 5√2 cm and one leg 5 cm.

Solution:

Draw one leg = 5 cm.
Construct 90° at one end.
From the other end, draw arc of radius 5√2 ≈ 7.07 cm.
The other leg = √((5√2)² − 5²) = √(50 − 25) = √25 = 5 cm.

Answer: Isosceles right triangle with legs 5 cm each.

Real-World Applications

Real-world uses:

Construction: Building right-angled corners with specific measurements.
Carpentry: Making right-angled joints with given dimensions.

Key Points to Remember

RHS construction needs a right angle, the hypotenuse, and one leg.
Hypotenuse must be longer than the given leg.
Construct the right angle first, then use an arc for the hypotenuse.
The triangle is unique.

Practice Problems

Construct △PQR: ∠Q = 90°, PR = 10 cm, QR = 6 cm.
Construct a right triangle: hypotenuse = 13 cm, one leg = 12 cm.
Can you construct a right triangle with hypotenuse 3 cm and one leg 5 cm? Why?

Frequently Asked Questions

Q1. What is RHS construction?

Constructing a right triangle given the right angle, hypotenuse, and one leg.

Q2. Why must the hypotenuse be longer than the leg?

By Pythagoras Theorem, hypotenuse² = leg₁² + leg₂². Since leg₂² > 0, the hypotenuse must be longer than each leg.

Q3. What tools are needed?

Ruler, protractor or set square (for 90°), and compass (for the hypotenuse arc).