Right Angled Triangle Constructions(RHS)
A triangle is a closed shape that has 3 sides. A right-angled triangle is a type of triangle having the 3 sides, named as the 'base', 'height', and the 'hypotenuse'. Right-angled triangles are extensively used in trigonometry. A right-angled triangle has one angle equal to 90° and the other two angles as acute. The side that is opposite to the right angle is called the 'hypotenuse', which is also the longest side of the right-angled triangle. The side that is adjacent to the right angle is called the 'adjacent side'. The height of the triangle forms the 'opposite side'.
Right Angled Triangle Definition
A triangle that has one of its internal angles equal to 90° is called a right-angled triangle. The longest side of a right-angled triangle is called the hypotenuse. The length of the longest side of the right-angled triangle can be found using the pythagoras theorem, which states that 'The sum of the square of the hypotenuse is equal to the sum of the squares of the other two sides.'
Right Angled Triangle Properties
The following points are the properties of a right-angled triangle.
- A right-angle triangle has one of its angles equal to 90°
- The longest side of a right-angle triangle is called 'Hypotenuse'.
- The sum of interior angles of a right-angled triangle is equal to 180°
- The side that is opposite to the right angle is called the opposite side and the side that is next to the right angle is the adjacent side.
- The area of a right-angled triangle is equal to (1/2 × base × height)
- If any two sides of a right-angled triangle are given we can construct the triangle easily.
- One of the unknown sides of a right-angled triangle can be found using the pythagorean theorem formula - Hypotenuse = √Opposite side2 + Adjacent side2
Construction of a Right Angled Triangle
A triangle in which one of the angles is equal to 90° is a right-angled triangle. The side which is directly opposite to the right angle is called the hypotenuse of the longest side. To construct a right-angled triangle, we require the measurements of two of its sides. A compass and a ruler are necessary to construct a right-angled triangle. Now let us see how a triangle PQR with the hypotenuse13 units and one of its sides to be 5 units.
Step 1: Draw a horizontal line and mark a point Q on it. The line can be of any length.
Step 2: With Q as the center, measure 5 units in a compass and draw an arc on both the sides of the point such that the arc touches the horizontal line and mark the points as 'S' and 'R'.
Step 3: With 'S' as the center and measuring 13 units in the compass draw an arc above 'S'.
Step 4: With the same width of 13 units, draw an arc from the point 'R'. Mark the point of intersection of these arcs as 'P'.
Step 5: Join the points 'P' and 'Q' and 'P' and 'R' with a ruler.
Step 6: The angle at point Q is 90°.
Topics Related to Right Angled Triangle Constructions
Check out some interesting topics related to right-angled triangle constructions.
Solved Examples on Right Angled Triangle Constructions
Example 1: Construct a right triangle ABC with AC = 5 units and BC = 3 units.
Follow the steps below to construct a right triangle ABC, AC = 5 units, and BC = 3 units.
- Step 1: Draw a horizontal line and mark a point 'B' on it.
- Step 2: With 'B' as the center, and measuring 3 units as width in a compass, draw two arcs on either side of the line and mark them as 'D' and 'C'.
- Step 3: With 'D" as the center and measuring 5 units as width in a compass, draw an arc above 'B' and mark it as 'A'.
- Step 4: Repeat the same process with 'C' as the center.
- Step 5: Join the point of intersection of these arcs with 'B' and 'C".
- Step 6: Mark angle B as 90°.
Please refer to the figure below, to see how the right-angled triangle will look like.
Example 2: The measurements of two sides of a right-angled triangle are as follows. Hypotenuse = 10 units and the opposite side = 6 units. how can you find the third side apart from the construction method that is using a compass and ruler?
We can do that using Pythagoras theorem, Hypotenuse2 = Opposite side2 + Adjacent side2. To find the adjacent side, we can rewrite the formula as follows, Adjacent side = √ Hypotenuse2 - Opposite side2. The side of the hypotenuse is 10 units and the opposite side is 6 units. Substituting the values in the formula we get, Adjacent side = √ 102 - 62 = √ 100 - 36 = √ 100 - 36 = √ 64 = 8 units. Therefore, the length of the adjacent side of the triangle is 8 units.
Practice Questions on Right Angled Triangle Constructions
FAQs on Right Angled Triangle Constructions
What Is a Right Angled Triangle?
A right-angled triangle is a type of triangle in which one of its interior angles is equal to 90°. The side opposite to the longest side is called the hypotenuse or the longest side.
What Are The Sides of a Right Triangle Called?
The hypotenuse is the longest side of the right-angled triangle, the side opposite to the right-angled triangle is the opposite side and the side that is next to the right angle is called the adjacent side.
What Do the Angles of a Right Triangle Add Up To?
The angles of the right-angled triangle add up to 180°. (add about one angle being 90 degrees and other two are acute, bt sum upto 180)
How to Find the Area of a Right Angle Triangle?
To find the area of a right triangle, check if base and height are given, then apply the formula = 1/2 x Base x height.
What Are the Properties of a Right Angled Triangle?
Some important properties of a right-angled triangle are as given below:
- One of its angles of a right angle is always equal to 90°.
- The sum of interior angles of a right angle is equal to 180°.
- The longest side of a right-angled triangle is called the hypotenuse.