Overview of Triangles
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 Triangles
1. How to Draw a Triangle When its Perimeter and Two Sides are Given. Mention its Steps of Construction.
For construction a triangle when its perimeter and two sides are given, you should have a ruler, a compass and protractor.
Let us take a triangle PQR whose length is 15 cm and two given angles are 55° and 60°.
Hence , P + Q+ R = 15 CM ,∠Q = 55° and ∠R = 65°
Steps of Construction
Draw a line segment XY of length AB +BC+AC = 15 cm
Using the protractor, construct an ∠LXY from the point Which is equivalent to ∠B =55°
Using the protractor, construct an ∠MYX from the point Which is equivalent to ∠C =60°
Construct the bisectors of ∠LXY and ∠MYX using the compass. Mark point A where two bisectors meet.
Draw the perpendicular bisector of AX and give a name to it as PQ.
Draw the perpendicular bisector of AY and give a name to it as RS.
Let PQ and RS touch the line segment XY at point B and C respectively.
Join the points A and B along with A and C.
2. Explain the Term Congruence of Triangle
The term congruence is used to define the object and the mirror image of any of the objects. Two shapes or objects will be considered as congruent if they overlap each other. The shapes and dimensions of the congruent triangle are similar. In Geometrical figures, the line segments within the same length and angle with the same measurement are equal.
Congruent Triangle
Congruent triangles are triangles having their sides and angles equal. The congruence is symbolized by the symbol ≅. The area and the perimeter of the congruent triangle are similar.
Two triangles are said to be congruent when their sides and angles have the same measure. Hence, two triangles can overlap side to side and angle to angle.
The triangle ABC and PQR, given below are congruent. It implies that,
Vertices- A and P , B and Q and C and R are similar.
Sides= AB=PR, QR=BC and AC=PR
Angles = ∠A = ∠P, = ∠Q AND ∠ C = ∠ R