How to Construct a Right Angled Triangle: Methods & Solved Problems
A simple closed curve or a polygon formed by three line-segments (sides) is known as a triangle.
A triangle has the following,
three line-segments or three sides
three vertices (vertex points)
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
A triangle that has all the three line-segments or sides unequal is known as a scalene triangle.
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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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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.
FAQs on Right Angled Triangle Constructions: Stepwise Guide & Practice
1. What are the essential conditions required to construct a unique right-angled triangle?
To construct a unique right-angled triangle, you don't need all three sides and angles. You only need a specific combination of information. The two most common conditions taught in the CBSE syllabus are:
- The lengths of the two legs (the sides that form the 90° angle).
- The length of the hypotenuse and one leg. This is known as the RHS (Right angle-Hypotenuse-Side) criterion.
2. How do you construct a right-angled triangle step-by-step when the lengths of its two legs are given?
Here is a simple, step-by-step guide to construct a right-angled triangle if you know the lengths of the two sides forming the right angle (the legs):
- Draw a line segment, say AB, which will be the base of the triangle, equal to the length of one of the given legs.
- At one point (e.g., A), use a protractor or a compass to construct a 90° angle. Let this perpendicular line be AX.
- Set your compass to the length of the second given leg. Place the compass point at A and draw an arc that intersects the perpendicular line AX. Mark this intersection point as C.
- Join points B and C with a straight line. The triangle ABC is the required right-angled triangle, where BC is the hypotenuse.
3. What is the RHS criterion, and how is it used for constructing a right-angled triangle?
The RHS criterion stands for Right angle, Hypotenuse, Side. It is a specific rule for constructing a right-angled triangle when you know the length of one side (a leg) and the length of the hypotenuse. The steps are:
- Draw a line segment that will serve as the base, equal to the length of the given leg. Let's call it QR.
- At one point (say, Q), construct a perpendicular line (a 90° angle).
- Set your compass to the length of the given hypotenuse.
- Place the compass point on the other end of the base (R) and draw an arc that cuts the perpendicular line. Name the intersection point P.
- Join P and R. The resulting triangle PQR is the required right-angled triangle.
4. How is the Pythagorean theorem related to the construction of a right-angled triangle?
The Pythagorean theorem (a² + b² = c²) is the fundamental principle that governs the relationship between the sides of a right-angled triangle. While it's not a construction tool itself, it plays two key roles:
- Verification: After constructing a triangle, you can measure its three sides to check if they satisfy the theorem. If they do, your 90° angle construction is accurate.
- Foundation for Methods: Practical construction techniques, like the '3-4-5 rule' used in construction and carpentry, are direct applications of Pythagorean triples to create a perfect right angle without a protractor.
5. What is the most common mistake made during construction using the RHS criterion?
A very common mistake when using the RHS criterion is placing the compass point incorrectly to draw the hypotenuse arc. Students often place the compass on the vertex where the 90° angle is constructed. This is wrong. The compass, set to the length of the hypotenuse, must be placed on the *other* end of the base leg (the leg whose length is known). The arc should then intersect the perpendicular line to find the third vertex of the triangle.
6. Is it possible to construct a unique right-angled triangle if you only know the length of the hypotenuse?
No, it is not possible to construct a unique right-angled triangle with only the length of the hypotenuse. Knowing the hypotenuse defines the diameter of a circle on which the right-angle vertex must lie. You can form an infinite number of different right-angled triangles with the same hypotenuse but with varying leg lengths. To make the triangle unique, you need one more piece of information, such as the length of one leg or the measure of one of the acute angles.
7. How is constructing a right-angled triangle different from constructing a general triangle (like a scalene triangle)?
The key difference lies in the given information. For a general triangle, you typically need three pieces of information, such as Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA). However, in a right-angled triangle, one piece of information is already known: one angle is always 90°. This reduces the amount of new information required. This is why you can construct it with just two sides (either two legs, or a leg and the hypotenuse under the RHS criterion), which wouldn't be sufficient for a general triangle.
