How to Construct a Right Angled Triangle with Given Sides and Hypotenuse
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 Explained Clearly
1. What is a right angled triangle in geometry?
A right angled triangle is a triangle that has one angle equal to 90°. In this triangle:
- The side opposite the 90° angle is called the hypotenuse.
- The other two sides are called the legs or perpendicular sides.
- The Pythagoras theorem applies only to right angled triangles.
2. How do you construct a right angled triangle using a ruler and compass?
To construct a right angled triangle, draw one side and erect a perpendicular at one endpoint using a compass. Steps:
- Draw a line segment AB.
- At point A, construct a 90° angle using a compass.
- Mark point C on the perpendicular line.
- Join B to C to form triangle ABC.
3. What is the formula for a right angled triangle?
The main formula for a right angled triangle is the Pythagoras theorem: c² = a² + b². Here:
- c = hypotenuse
- a and b = perpendicular sides
4. How do you construct a right angled triangle when the hypotenuse is given?
To construct a right angled triangle with a given hypotenuse, use the semicircle method. Steps:
- Draw the given hypotenuse AB.
- Find the midpoint of AB.
- Draw a semicircle with AB as diameter.
- Choose any point C on the semicircle and join AC and BC.
5. How do you construct a right angled triangle with given base and height?
To construct a right angled triangle with given base and height, draw the base and erect a perpendicular equal to the height. Steps:
- Draw base AB of given length.
- At point A, construct a 90° angle.
- Mark point C such that AC equals the given height.
- Join BC.
6. What are the properties of a right angled triangle?
A right angled triangle has one angle equal to 90° and follows specific geometric properties. Key properties:
- The side opposite 90° is the longest side (hypotenuse).
- Pythagoras theorem applies.
- The two acute angles add up to 90°.
- The area is calculated as ½ × base × height.
7. How do you find the area of a right angled triangle?
The area of a right angled triangle is calculated using Area = ½ × base × height. Since the base and height are perpendicular:
- If base = 8 cm and height = 6 cm,
- Area = ½ × 8 × 6 = 24 cm².
8. Can you give an example of constructing a right angled triangle?
Yes, for example, construct a right angled triangle with sides 3 cm and 4 cm. Steps:
- Draw AB = 4 cm.
- At A, construct a 90° angle.
- Mark AC = 3 cm on the perpendicular.
- Join BC.
9. What is the difference between a right angled triangle and an acute triangle?
A right angled triangle has one angle equal to 90°, while an acute triangle has all angles less than 90°. Differences:
- Right triangle follows c² = a² + b².
- Acute triangle does not have a 90° angle.
- The longest side in a right triangle is opposite the 90° angle.
10. What are common mistakes in right angled triangle constructions?
Common mistakes in right angled triangle constructions include incorrect perpendicular drawing and measurement errors. Frequent errors:
- Not constructing an exact 90° angle.
- Confusing the hypotenuse with a leg.
- Applying Pythagoras theorem incorrectly.
- Using inaccurate compass width while drawing arcs.
