Written By Keerthi Kulkarni
Last Modified 25-01-2023

Construction of Right-Angled Triangle Using RHS Criterion

A triangle is a three-sided closed shape. A right-angled triangle has three sides, referred to as the ‘base,’ ‘height,’ and ‘hypotenuse.’ Generally, the construction of a right-angled triangle is done using RHS Criterion.

One of the angles in a right-angled triangle is 90 degrees, while the other two are sharp. The ‘hypotenuse,’ the longest side of the right-angled triangle, is the side opposite the right angle. This article let us understand how to construct a right-angled triangle using the RHS rule.

What is a Right Triangle?

A triangle is a three-sided closed polygon. There are different triangles based on the sides and angles of the triangle, such as equilateral, isosceles, acute-angled, obtuse-angled and right-angled triangles etc.

A triangle in which one angle equals the right angle \(\left( {90^\circ } \right)\) is called the right-angled triangle. Generally, a right-angled triangle has three sides known as base, altitude and hypotenuse. The largest side or the side opposite to the right angle in the right-angled triangle is called the hypotenuse.

A right triangle, in which the lengths of the base and the altitudes are equal, is said to be a right-angled isosceles triangle. The other two angles in a right-angled isosceles triangle are equal, and each being \(45^\circ \).

Properties of Right Triangle

The properties of the right-angled triangles are listed below:

One of the angles in the right triangle is a right angle \(\left(90^{\circ}\right)\).
The longest or largest side of the right triangle is called the hypotenuse.
The hypotenuse lies opposite to the right angle.
The sum of the other two angles equals the right angle \(\left(90^{\circ}\right)\).
The perpendicular drawn from the right angle to the hypotenuse divides the two triangles into similar triangles.
The other two angles in a right-angled isosceles triangle are equal, and each being \(45^{\circ}\)
The length of the hypotenuse is found by using Pythagoras theorem as \(\text {hyp}^{2}=\operatorname{side}^{2}+\operatorname{side}^{2}\)

Pythagoras Theorem

The Pythagoras theorem is used to find the length of the hypotenuse. It states that the square of the hypotenuse equals the sum of the squares of the legs (sides) of the right triangle.

\(\operatorname{Hyp}^{2}=\operatorname{Side}^{2}+\operatorname{Side}^{2}\)

In a right triangle of the sides \(a, b, c\), then the length of the hypotenuse is given by
\(c^{2}=a^{2}+b^{2}\).

RHS Congruence Rule

Two triangles are said to be congruent if they have the same shape and same size. The congruence rule compares two triangles’ three measurements (sides or angles). We have different congruency rules such as SSS, SAS, ASA and RHS.

RHS congruence rule is used to compare whether the two right triangles are congruent or not. RHS congruence rule is also used to construct the right-angled triangle.

RHS congruence rule states that if the side of one right-angled triangle is equal to the hypotenuse and a side of the other right-angled triangle, then those two right triangles are congruent. RHS criterion defines the \(\text {right angle}-\text {hypotenuse}-\text {side}\) congruence criterion.

In the above-given triangles \(\triangle A B C, \Delta E F G\),

\(A C=E F\) (Represented by a single line).
\(B C=F G\) (Represented by two lines).
\(\angle A B C=\angle E G F\) (Right angles)

Here, sides \((BC, AC)\) and right angle \(\angle A B C\) of triangle \(A B C\) are equivalent to the corresponding sides \((F G, E F)\) and the right angle \(\angle F G E\) of the triangle \(E F G\).

\(\triangle A B C \cong \Delta E G F\)

Tools Used to Construct the Triangle

Geometry is the branch of mathematics that deals with studying shapes or objects and their construction. In geometry, we have different shapes or objects to be drawn or constructed using various tools. Some of the frequently used tools are discussed here:

1. Scale: A scale or ruler is used to draw the length of the sides of the given triangle. Put the pencil at the zero mark of the scale and draw the line segment of the given length by using the markings on the scale.

Example: Drawing length of \(4 \,\text {units}\) by using a ruler as given below:

2. Compass: It is used to cut the arc of any given length. Place the needle of the compass at a point and mark the arc at a line with the given length.

3. Protractor: A protractor is the tool that is used to construct angles. Place the centre of the protractor at the point and mark the given angle.

Construction of Right-Angled Triangle using RHS Criterion

Let us discuss the construction of the triangle with the help of an example when the measurements of the side and hypotenuse of the right-angled triangle are given.

Example: Construct a right-angled triangle with sides \(A B=3 \mathrm{~cm}, B C=5 \mathrm{~cm}\) and \(\angle A=90^{\circ}\).

Step-1: Draw a line \(AB\) of length \(3 \,\text {cm}\) by using the scale and pencil.

Step-2: Draw an angle of \(90^\circ \) from point \(A\) by using a protractor.

Step-3: Draw an arc of length \(5 \,\text {cm}\) from point \(B\) by using a compass, which cuts the previous line at \(C\).

Step-4: Join \(B C\). Thus, \(\triangle A B C\) is the required triangle.

The above-drawn triangle \(ABC\) is the right-angled triangle.

Construction of Right-Angled Isosceles Triangle Using RHS Congruence Rule:

A right triangle, in which the lengths of the base and the altitudes are equal, is said to be a right-angled isosceles triangle. Let us discuss the construction of the right-angled isosceles triangle by taking the example:

Let us construct the right-angled isosceles triangle of sides other than hypotenuse equal to \(4\) units.

Step-1: Draw a line \(QR=4 \,\text {units}\) with scale and pencil.

Step-2: At point \(Q\), draw an angle of \(90^\circ \) with protractor.

Step-3: At point \(Q\), draw an arc of length \(PQ=4 \,\text {units}\) by compass.

Step-4: Join \(PR\). Thus, triangle \(PQR\) is the required one.

Solved Examples

Q.1. Construction of right-angled triangle using RHS criterion \(ABC\), in which \(AB=5 \,\text {cm}\) and \(BC=13 \,\text {cm}\) and right-angled at \(A\).
Ans: Step-1: Draw a line \(A B\) of length \(5 \mathrm{~cm}\) by using the scale and pencil.
Step-2: Draw an angle of \(90^{\circ}\) from point \(A\) using a protractor.
Step-3: Draw an arc of length \(13 \mathrm{~cm}\) from point \(B\) by using a compass, which cuts the previous line at \(C\).
Step-4: Join \(B C\)
Step-5: Thus, \(\Delta A B C\) is the required triangle.

The above-drawn triangle \(ABC\) is the right-angled triangle.

Q.2. Construct the right-angled isosceles triangle \(PQR\), right-angled at \(Q\) and \(PQ=QR=7\) units.
Ans: Step-1: Draw a line \(Q R=7\) units with scale and pencil.
Step-2: At point \(Q\), draw an angle of \(90^{\circ}\) with protractor.
Step-3: At point \(Q\), draw an arc of length \(Q P=7\) units by compass.
Step-4: Join \(P R\). Thus, triangle \(P Q R\) is the required one.

Q.3. Construct the right-angled triangle \(ABC\), right-angled at \(B\). \(BC=8 \,\text {cm}, AC=10\,\text {cm}\).
Ans: Step-1: Draw a line segment \(BC=8 \,\text {cm}\).

Step-2: Draw or construct the right angle at point \(B\) by using the protractor.
Step-3: From point \(C\), draw an arc of length \(10 \,\text {cm}\), cuts the ray \(BX\) at \(A\).

Step-4: Join the points \(A, C\).
Step-5: Thus, the triangle \(ABC\) is the required right-angled triangle.

Q.4. Construct a triangle \(A B C\) congruent to the right triangle \(X Y Z\),in which \(Y Z=7 \mathrm{~cm}, X Y=5 \mathrm{~cm}\) and \(\angle Y=90^{\circ}\).
Ans: Given \(\triangle A B C \cong \triangle X Y Z\),
So, \(Y Z=B C=7 \mathrm{~cm}, X Y=A B=5 \mathrm{~cm}\) and \(\angle Y=\angle B=90^{\circ}\)
Step-1: Draw a line segment \(B C=5 \mathrm{~cm}\).
Step-2: Construct the right angle at point \(B\) by using the protractor.
Step-3: From point \(C\), draw an arc of \(7 \mathrm{~cm}\), and cut the ray at point \(A\).
Step-4: Join points \(A, C\).
Step-5: Thus, the triangle \(A B C\) is the required right triangle that is congruent to the triangle \(X Y Z\).

Q.5. Construct the right-triangle \(PQR\) congruent to \(ABC\), in which \(AB=BC=4\) units.
Ans: Given \(\Delta P Q R \cong \Delta A B C, P Q=Q R=A B=B C=4\) units.
Step-1: Draw a line \(Q R=4\) units with scale and pencil.
Step-2: At point \(Q\), draw an angle of \(90^{\circ}\) with protractor.
Step-3: At point \(Q\), draw an arc of length \(P Q=4\) units by compass.
Step-4: Join \(P R\). Thus, triangle \(P Q R\) is the required one.

Summary

In this article, we have studied the definition of the right-angled triangle and right-angled isosceles triangle and the properties of the right triangles. We have discussed the RHS congruence rule. This article gives the construction of right-angled triangles and right-angled isosceles triangles by using the RHS congruence rule.

We have discussed the geometrical tools to construct the triangles, such as ruler, compass and protractor. This article gives the solved examples of the construction of right triangles that help us understand the concepts easily.

FAQs

Q.1. What is the RHS rule in a triangle?
Ans: RHS congruence rule states that if the hypotenuse and a side of one right-angled triangle are equal to the hypotenuse and a side of the other right-angled triangle, then those two right triangles are congruent to each other.

Q.2. Which of the following conditions must be given for constructing a triangle by RHS criterion?
Ans: The conditions given for constructing the triangle by RHS criterion are
1. Hypotenuse
2. Any one side of the triangle

Q.3. Which congruency rule is used to construct the right-angled triangle?
Ans: The R.H.S congruency rule is used to construct the right-angled triangle.

Q.4. What is RHS stands for?
Ans: RHS stands for right angle-hypotenuse-side.

Q.5. What condition must be followed to construct a triangle by using RHS criterion?
Ans: The condition to be followed is that the length of the hypotenuse and any side of the right-angled triangle must be given.