Right Angled Triangle Constructions

A simple closed curve or a polygon formed by three line-segments (sides) is known as a triangle.

A triangle has the following,

1. three line-segments or three sides

2. three vertices (vertex points)

3. three angles

What are the Types of Triangles?

There are six types of triangles, three with respect to sides and three with respect to angles.

Three Types of Triangle with Respect to Sides of a Triangle

1.  A triangle that has all the three line-segments or sides unequal is known as a scalene triangle.

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2.  A triangle has a pair of its sides or two line-segments equal is called an isosceles triangle.In this case, AB equals AC.

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3.  A triangle having all the three line-segments or sides equal is known as an equilateral triangle. Here AB equals BC equals CA.

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Types of Triangle with Respect to Angles

(i) A triangle in which all the three angles are acute is known as an acute angled triangle. Angle ABC, ∠ACB and BAC are all acute angles.

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(ii) A triangle in which one of the three angles is a right angle is known as a right angled triangle. Here , angle ABC = one right angle.

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(iii) A triangle where one of the three angles is more than a right angle (or is an obtuse angle) is known as obtuse angled triangle. Here, angle ABC is an obtuse angle.

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What is a Right Angle Triangle?

Consider a right-angle triangle ABC, with its three sides namely the opposite, adjacent and the hypotenuse. In a right-angled triangle we generally refer to the three sides in order to their relation with the angle . The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.

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Right Angle Triangle Properties

We will discuss the properties of a right angle triangle.

• One angle is always equal to 90° or the right angle.

• The side opposite angle is equal to 90° is the hypotenuse.

• The hypotenuse is always the longest side in a triangle.

• The sum of the other two interior angles of a triangle is equal to 90 degrees.

• The other two sides adjacent to the right angle are known as the base(B) and perpendicular(P).

• The area of right angle triangle is equal to half of the product of the two adjacent sides of the right angle, that is,

Right Angle Triangle Area = ½ (B × P)

• If we drop a perpendicular from the right angle to the hypotenuse, we get three similar triangles.

• If we draw a circumcircle which passes through all three vertices, then the radius of this circle drawn by us is equal to half of the length of the hypotenuse.

• If one of the angles is equal to 90° and the other two angles are equal to 45 degrees each, then the triangle is known as an Isosceles Right Angled Triangle, where the adjacent sides to 90 degrees are equal in length to each other.

This Criteria for construction of the Triangle is possible when the Hypotenuse and one We need a ruler and a compass for the construction of a triangle. We will construct a right-angled triangle ABC, and the right angle is at C. You need to consider the length of the hypotenuse AB equals 5 cm and side CA equals 3 cm. The construction rhs steps are given below.We are going to discuss right angled triangle constructions rhs below.

Constructions rhs Steps

• Step 1: First, you need to draw a horizontal line of any length and mark a point name C on it.

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• Step 2: Now set the compass width to 3 cm.

• Step 3: After doing that now place the pointer head of the compass on the point C and you need to mark an arc on both the sides of C.

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• Step 4: Mark the two points as P and A where the arcs cross the line.

• Step 5: Now set the compass width to the length of the hypotenuse, which is the longest side, that is, equal to 5 cm.

• Step 6: Place the pointer head of the compass on the point P and then mark an arc above C.

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• Step 7: Repeat step 6 from point A.

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• Step 8: Mark the point you get as B where the two arcs cross each other.

• Step 9: Join the point B and point A as well as point B and point C with the ruler.

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Questions to be solved

Question 1) How will you construct a right angled triangle?

Answer)Constructions rhs steps:

• Step 1: First, you need to draw a horizontal line of any length and mark a point name C on it.

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• Step 2: Now set the compass width to 3 cm.

• Step 3: After doing that now place the pointer head of the compass on the point C and you need to mark an arc on both the sides of C.

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• Step 4: Mark the two points as P and A where the arcs cross the line.

• Step 5: Now set the compass width to the length of the hypotenuse, which is the longest side, that is, equal to 5 cm.

• Step 6: Place the pointer head of the compass on the point P and then mark an arc above C.

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• Step 7: Repeat step 6 from point A.

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• Step 8: Mark the point you get as B where the two arcs cross each other.

• Step 9: Join the point B and point A as well as point B and point C with the ruler.

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Practice Questions for You!

1) If the sides of the triangle are given as 3cm,4cm and 5cm then what type of triangle will you get after constructing it?

2) Construct a right angled triangle named ABC where AB = 4.5 cm, AC = 5.8 cm and  the angle A = 90 degrees.

3) In a triangle PQR, if ∠P=90 degrees and ∠Q= ∠R, find the angles of the triangle.