Introduction- Constructing Triangles
Three line segments combine to form a triangle (a polynomial shape), resulting in three sides and three angles. The triangle can be made with geometric tools such as a ruler, protractor, and compass.
A triangle is formed when 3 line sections create a closed curve or polygon. The three main parts found in a triangle are its vertices, angles, and sides. We will learn how to build several types of triangles in this lesson. For more information, keep reading.
Different Variations of Triangle
Having 3 angles, sides, and vertices, a triangle is a three-dimensional polygon. Triangles come in a variety of shapes depending on their sides, angles, and other characteristics.
- Equilateral Triangle: When a triangle has three equal sides and three equal angles then it is called an equilateral triangle.
- Isosceles Triangle: When a triangle has two equal sides then it is called an isosceles triangle.
- Scalene Triangle: When a triangle has no equal side then it is called a scalene triangle.
Triangles based on angles
- Acute-angled triangle, when all the angles are less than 90°.
- Right-angled triangle, when one angle is a right angle of 90°.
- Obtuse-angled triangle, when one of the angles is more than 90°.
Triangle Rules and their Properties
- PQR stands for a triangle with the vertices P, Q, and R.
- By the angle sum property, the sum of all angles in a triangle is 180°.
- In the triangle, the sum of the lengths of any two sides is always greater than the length of the third side.
- The difference in length between any two sides of the triangle is always lesser than the length of the third side.
- The sum of two opposing internal angles forms a triangle’s outer angle.
- In a triangle, the side that faces the angle’s largest side is the largest.
Triangle Construction – SSS
An illustration is used to demonstrate how a triangle is constructed when three sides are given:
Create a triangle, ABC for instance, with sides of 7cm, 3cm, and 5cm.
Step 1: First, draw a line with the length BC=5cm.
Step 2: From point B, create an arc with a compass so that BA equals 7cm.
Step 3: Use a compass to draw another arc from point C with a length of CA=3cm.
Step 4: Draw the line from other points (B, C) to the junction point (A).
△ABC is the appropriate triangle.
Triangle Construction – ASA
An illustration is used to demonstrate how a triangle is constructed when two angles and one side are given:
Create a triangle, PQR for instance, with a side of PQ=6cm, P=30°, and Q=60°.
Step 1: Using a pencil and a scale, make a line PQ that is 6 cm long.
Step 2: Using the protractor, now draw an angle of 30° from point P.
Step 3: Using the protractor, draw a 60° angle starting at point Q.
Step 4: Mark the junction point as R in step 4 to construct the required triangle of PQR
△PQR is the appropriate triangle.
Triangle construction using SAS
Let’s learn about how a triangle is made when two side measurements and one angle are provided:
Create a triangle, ABC for instance, with sides AB=6 cm, AC=4 cm, and A=30°.
Step 1: Using the scale and pencil, draw a line AB that is 6 cm long.
Step 2: Using a protractor, draw a 30° angle from point A.
Step 3: Using a compass, draw an arc from point A that is 4 cm long and intersects the preceding line at C.
Step 4: Join up BC. Therefore, the necessary triangle is △ABC.
△ABC is the appropriate triangle.
Triangle Construction – RHS
When the side and hypotenuse measurements of a right-angled triangle are given, let’s talk about how the triangle is constructed.
Create a triangle, for instance, with sides AB=4 cm, BC=7 cm, and A=90.
Step 1: Using the scale and pencil, draw a line AB that is 4 cm long.
Step 2: Using a protractor, draw a 90-degree angle starting from point A.
Step 3: Using a compass, draw an arc from point B that is 7 cm long and intersects the preceding line at C.
Step 4: Join the points BC. Therefore, the necessary triangle is △ABC.
△ABC is the appropriate triangle.
Different cases involving the construction of triangles
Construction of a triangle where the difference of the lengths of the other 2 sides are given along with the base and the base angle.
Make a triangle ABC using the following measurements: BC=8 cm, B=45°, and AB-AC=3.5 cm.
Construction steps include:
Step 1: Draw the segment of a line BC=10 cm.
Step 2: Draw an angle CBO=45 at point B using the compass.
Step 3: Cut the previous line with a 3.5 cm arc starting at point B.
Step 4: Join the points DC.
Step 5: Make a line drawing of the perpendicular bisector LM of CD that crosses BO at A.
Step 6: Join AC.
Therefore, the necessary triangle is the △ABC.
We can precisely create a triangle if we know the values of the sides and angles.
Construction of a triangle, when two base angles and the total of the 3 sides (perimeter) are combined.
For instance, build the triangle PQR with the angles P=30°, R=90°, and PQ+QR+RP=11cm.
Construction Steps:
- Using the scale and a pencil, draw a line segment PQ=11 cm.
- Draw an angle QPA=30 at point A using a compass.
- B uses the compass to create an angle PQB=90 at a specific point.
- The bisectors of QPA and PQB, which meet at Q, should be drawn.
- Join AY and BY
- Make both LM and ST perpendicular bisectors of AQ and BQ, respectively.
- At P and R, respectively, LM and ST cross AB.
- Now join PQ and QR.
- The required triangle is therefore the triangle △PQR.
Constructing Triangles FAQS:
Q1. What rule is applied in the construction of a triangle?
Ans. A few standards upon which angles are created are listed below:
Triangle with known sides according to the SSS criterion.
A triangle with two known angles and one known side meets the ASA criterion.
Triangle with two known sides and one known included angle, according to the SAS criterion.
Triangles that meet the RHS requirement have known hypotenuses and other sides.
Q2. How to make a three-sided triangle?
Ans. With the length of the longest side, draw a line. Draw two arcs so that they cross each other from the line’s two endpoints. Connect the longest side’s vertices to the intersection point.
Q3. How to correctly construct a triangle?
Ans. To construct a triangle, you can follow these steps:
- Draw a straight line segment and mark the desired length of one side of the triangle.
- Use a protractor to measure and mark the desired angles of the triangle at each end of the side.
- Connect the marked points with straight line segments to complete the triangle.
Q4. Enumerate the geometrical instruments that go into making a triangle.
Ans. Compass, protractor, ruler, etc. are some of the geometrical instruments used in the construction of triangles.
Q5. What are some applications for triangle construction?
Ans.
- Surveying: Triangle construction is used in surveying to measure distances and angles on the Earth’s surface.
- Engineering: Triangle construction is used in engineering to design structures, such as bridges and buildings, that can withstand various loads and forces.
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