Geometry 1 Triangle related post
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Construction of Triangles
A triangle is a two-dimensional closed shape with three sides. In order to construct a triangle, we should know the measure of the length of its sides and angles. But how do we construct them? Here comes in the concept of 'Geometry'. Geometry is a branch of math that deals with constructing figures like triangles, squares, circles, and many other shapes. These figures can be constructed with the help of geometrical instruments like ruler, protractor, or compass. Now let us read more about the construction of triangles.
How are Triangles Constructed?
Triangles can be constructed using a ruler and a compass and even with the help of a protractor. Triangles can be classified based on their sides and angles. We will discuss the steps, properties, and criteria to construct various triangles in the following sections.
Keeping the following properties of triangles in mind, triangles can be easily constructed.
- A triangle has three sides, three vertices, and three angles.
- The sum of interior angles of a triangle is equal to 180°. This property is called the angle sum property of a triangle.
- All the sides of a triangle may or may not be equal.
- All the angles of a triangle may or may not be equal.
- A triangle with vertices A, B, C is denoted as triangle ABC.
Types of Construction of Triangles
Triangles can be classified based on angles and sides. Similarly, the construction of triangles can also be classified as:
- Construction of Triangles Based on Sides
- Construction of Triangles Based on Angles
Let us read about the two types of construction of triangles in detail.
Construction of Triangles Based on Sides
Based on the properties of a triangle, all the sides of a triangle may or may not be equal, therefore, we need to construct three different types of triangles.
- An equilateral triangle in which all its sides are equal.
- An isosceles triangle in which two sides are equal.
- A scalene triangle in which there are three unequal sides.
Let us learn how to construct these three types of triangles.
Construction of Equilateral Triangle
To construct an equilateral triangle we need to remember that side 1 = side 2 = side 3. Let us construct an equilateral triangle.
Example: Draw an equilateral triangle XYZ with the sides of the triangle equal to 6 units each.
Solution: An equilateral triangle has all its sides of equal length. Therefore, keeping this in mind we will use the following steps for the construction:
- Step 1: Draw a line segment YZ = 6 units.
- Step 2: Use a ruler and measure 6 units with the compass. With Y as the center draw an arc above the line YZ.
- Step 3: With Z as the center and without changing the measure of length taken in the compass draw another arc to intersect the previous arc.
- Step 4: Join the points XY and XZ to get an equilateral triangle XYZ.
Construction of Isosceles Triangle
To construct an isosceles triangle, we need to remember that side 1 = side 2, but side 3 is different. Let us construct an isosceles triangle.
Example: Draw an isosceles triangle ABC with two sides of the triangle equal to 6 units and one side equal to 5 units.
Solution:
An isosceles triangle has two equal sides and a different side. Therefore, keeping this in mind we will use the following steps for the construction:
- Step 1: Using a ruler and a pencil draw a line segment BC of length = 5 units.
- Step 2: Place your compass needle at B and draw an arc with a measure of 6 units above the line BC.
- Step 3: Now, place the needle of the compass at C and draw an arc with the same measure of 6 units such that the two arcs should intersect at point A.
- Step 4: Join the points AB and AC to form an isosceles triangle ABC.
Construction of Scalene Triangle
To construct a scalene triangle the condition we need to remember that side 1 ≠ side 2 ≠ side 3. Let us construct a scalene triangle.
Example: Draw a scalene triangle ABC with three sides of the triangle equal to 7 units, 5 units, and 6 units.
Solution:
In a scalene triangle, all the sides are of different lengths. Therefore, keeping this in mind we will use the following steps for the construction:
- Step 1: Draw a line segment BC which measures 7 units.
- Step 2: With point B as the center and taking a measure of 5 units in the compass draw an arc above the line BC.
- Step 3: With point C as the center and taking a measure of 6 units in the compass draw an arc to intersect the arc drawn in step 2.
- Step 4: Now join the points AB and AC to get a scalene triangle ABC.
Construction of Triangles Based on Angles
The angle sum property of a triangle states that the sum of interior angles of a triangle is equal to 180°. All the angles of a triangle may or may not be equal. With this, we can construct three different types of triangles.
- An acute triangle
- A right-angled triangle
- An obtuse angle
Let us learn how to construct triangles based on angles.
Construction of an Acute Triangle
To construct an acute triangle, we should remember that all its angles are acute angles. Let us construct an acute-angled triangle.
Example: Construct a triangle XYZ with the base as 8 units. ∠X = 45° and ∠Y = 65°
Solution: To construct an acute-angled triangle with the given dimensions we will use the following steps for the construction:
- Step 1: Use a ruler and draw a horizontal line of length 8 units. Name the endpoints of this line as XY.
- Step 2: Place the center of the protractor on X and look for 45° in the scale of the protractor and mark it as point Z.
- Step 3: Now place the center of the protractor on Y and look for 65° in the protractor.
- Step 4: Join XZ and YZ.
- Step 5: We have an acute-angled triangle with ∠X = 45°, ∠Y = 65° and ∠Z = 70°
Construction of a Right Angled Triangle
To construct a right-angled triangle, it should be remembered that the triangle must have one right angle. Let us construct a right-angled triangle.
Example: Construct a right-angled triangle PQR with one of its sides as 4 units.
Solution: To construct a right-angled triangle with the given dimensions we will use the following steps for the construction:
- Step 1: Draw a horizontal line QR of length 4 units.
- Step 2: Place the center of a protractor on Q and look for 90°. Mark the point as 'P'.
- Step 3: Join PQ and PR.
- Step 4: ∠PQR is 90° and triangle PQR is a right angled triangle.
Click on the link to know more about Right Angled Triangle Constructions (RHS)
Construction of an Obtuse Triangle
To construct an obtuse triangle we should remember that the triangle must have one obtuse angle. Let us construct an obtuse angle triangle.
Example: Construct a triangle XYZ with XY = 7 units, ∠X = 40°,∠Y = 105°.
Solution: To construct an obtuse triangle with the given dimensions we will use the following steps for the construction:
- Step 1: Draw a line segment with XY = 7units.
- Step 2: With X as the center, use a protractor to measure ∠X = 40° and draw a ray 'XP'.
- Step 3: With Y as the center, use a protractor to measure ∠Y = 105° and draw a ray 'YQ' such that it intersects with 'XP'. Mark the point of intersection as 'Z'.
- Step 4: Join the points ZX and ZY to make the triangle complete and form an obtuse triangle XYZ.
It should be noted that the third angle, which is ∠Z can be found using the angle sum property of triangles. So, ∠Z = 180 - (40 + 105) = 35°.
Construction of Triangles when Three Sides are Given
The construction of a triangle can be easily done with the help of a ruler and a compass when three sides are given, Let us understand the process with the help of an example.
Example: Construct a triangle PQR with the given sides: PQ = 5 units, QR = 6 units, and PR = 3.5 units
Solution: To construct a triangle with the above dimensions, we will use the following steps:
- Step 1: Draw a line segment QR measuring 6 units.
- Step 2: With Q as the center, take a measure of 5 units in the compass and draw an arc.
- Step 3: With R as the center, take a measure of 3.5 units in the compass and draw an arc intersecting the previous arc.
- Step 4: Connect the lines PQ and PR to form a triangle PQR.
Note: A triangle that is drawn with the lengths of all the three sides known is an SSS triangle.
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