How to Construct a Triangle Using SSS SAS ASA and RHS Criteria with Step by Step Explanation
A triangle is a three-sided polygon that has three edges and three vertices and the sum of all the three internal angles of any given triangle is 180°. To construct a triangle, geometrical tools are needed. Using a ruler, compasses, protractor and a pencil, a triangle can be constructed. For constructing a triangle, it is important to have the following dimensions:
All the three sides of a triangle
Two sides and one angle
Two angles and one side
The hypotenuse of a right-angled triangle along with another side of a triangle.
Before getting into the construction of a triangle, let's know what are the properties of a triangle to keep in mind while constructing a triangle.
Properties
There are many types of triangles such as equilateral triangle, scalene triangle, acute-angle triangle, isosceles triangle, obtuse-angled triangle, right-angled triangle, but all the triangles have some common properties:
The sum of all the internal angles of a triangle equals 180°.
The sum of two adjacent internal angles is equal to the external angle of the opposite side.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side of the triangle.
A right-angled triangle has a property called Pythagoras theorem which states that the square of the hypotenuse side of the triangle is equal to the sum of squares of the two other sides.
Construction
Based on the dimensions given for construction, they can be classified into three categories:
SSS - when three sides are given.
SAS - when one angle and two sides are given.
ASA - when two angles and one side are given.
Construction of SSS triangle
When three sides of a triangle are given, construction of a SSS triangle is possible using the following directions:
Draw a line segment of length equal to the longest side of the triangle.
Using a ruler, measure the length of the second side and draw an arc.
Then take the measurement of the third side and cut the previous arc and mark the point.
Now join the endpoints of the line segment to the point where the two arcs cut each other and get the required triangle.
Construction of SAS triangle
When two sides and an internal angle of a triangle is given then the SAS triangle can be constructed as follows:
Draw a line segment of length equal to the longest side of the triangle using a ruler and pencil.
Put the center of the protractor on one end of a line segment and measure the given angle. Join the points and construct a ray, such that the ray is nearer to the line segment.
Take measurement of the other given side of the triangle using a ruler and a compass.
Then put the compass at one end and cut the ray at another point.
Now join the other end of the line segment to the point.
Construction of ASA triangle
When two angles and a side are given, an ASA triangle can be constructed in the following way:
Draw a line segment of length equal to the given side of the triangle, using a ruler.
At one endpoint of a line, segments measure one of the given angles and draw a ray.
At another endpoint of the line segment, measure the other angle using a protractor and draw another ray such that it cuts the previous ray at a point.
Join the previous point with both the endpoints of the line segment and get the required triangle.
Construction of a Right-Angled Triangle
When the hypotenuse of a triangle is given along with the two other sides of the triangle, a right-angled triangle can be constructed as follows:
Draw the line segment equal to the measure of hypotenuse side
At one of the endpoints of the line segment, measure the angle equal to 90° and draw a ray
Then measure the length of another given side and draw an arc to cut the ray at a point and name it
Now join the point to the other side of the line segment to get the required right-angled triangle.
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FAQs on Construction of Triangle with Geometric Methods and Proof
1. What is construction of a triangle in geometry?
The construction of a triangle is the process of drawing a triangle accurately using given measurements with a ruler and compass. It involves constructing sides and angles based on specific data such as:
- Three sides (SSS)
- Two sides and included angle (SAS)
- Two angles and one side (ASA or AAS)
2. How do you construct a triangle when three sides are given (SSS)?
To construct a triangle with three given sides (SSS construction), draw one side and use arcs to locate the third vertex. Follow these steps:
- Draw base AB equal to one given side.
- With center A and radius equal to the second side, draw an arc.
- With center B and radius equal to the third side, draw another arc intersecting the first arc at point C.
- Join AC and BC to complete triangle ABC.
3. What is the triangle inequality condition in triangle construction?
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. Mathematically:
- a + b > c
- b + c > a
- c + a > b
4. How do you construct a triangle with two sides and the included angle (SAS)?
To construct a triangle using SAS (Side-Angle-Side), draw the given angle between the two known sides. Steps:
- Draw base AB equal to one given side.
- At point A, construct the given angle using a protractor.
- On the angle arm, mark point C such that AC equals the second given side.
- Join BC to complete the triangle.
5. How do you construct a triangle when two angles and one side are given (ASA)?
In ASA construction, draw the given side and construct the two given angles at its endpoints. Steps:
- Draw base AB equal to the given side.
- At A, construct the first given angle.
- At B, construct the second given angle.
- The two angle arms intersect at point C; join AC and BC.
6. Can a triangle be constructed if two sides and a non-included angle (SSA) are given?
A triangle with SSA (Side-Side-Angle) may result in zero, one, or two possible triangles, known as the ambiguous case. The outcome depends on the side lengths and the given angle:
- If the opposite side is too short, no triangle is formed.
- If conditions are exact, one triangle is formed.
- If measurements allow, two different triangles are possible.
7. How do you construct an equilateral triangle?
An equilateral triangle is constructed by drawing arcs of equal radius from both endpoints of a line segment. Steps:
- Draw line segment AB of given length.
- With center A and radius AB, draw an arc.
- With center B and radius AB, draw another arc intersecting at C.
- Join AC and BC.
8. How do you construct an isosceles triangle?
An isosceles triangle is constructed by making two equal sides from the endpoints of a base. Steps:
- Draw base AB.
- With center A and chosen radius, draw an arc.
- With center B and the same radius, draw another arc intersecting at C.
- Join AC and BC.
9. What instruments are required for triangle construction?
The basic instruments required for triangle construction are a ruler, compass, and protractor. These tools are used as follows:
- Ruler – to draw straight line segments.
- Compass – to draw arcs and circles with fixed radius.
- Protractor – to measure and construct given angles.
10. How do you know if a triangle construction is correct?
A triangle construction is correct if all given sides and angles match the required measurements and satisfy triangle properties. Verify by:
- Measuring each side to confirm given lengths.
- Checking angles with a protractor.
- Ensuring the angle sum equals 180°.
- Confirming the triangle inequality condition.
