Right angled triangle: Constructions (RHS)

Introduction

The word trigonometry means tri-angle-measurements which is three angle measurement. When we take any polygon, such as square, rectangle, pentagon, hexagon, etc., we can divide each polygon into triangles. So trigonometry mainly deals with triangles. There are three types of triangle based on the angle measurements. They are acute angled triangle, obtuse angled triangle, right angle triangle.

Acute angled triangle − All the three interior angles measured are less than 90°.

Obtuse angled triangle − All the three interior angles measured are greater than 90°.

Right angled triangle − At Least one of the interior angles measures 90°.

Right triangles

In trigonometry, the right angled triangle is more important than the other two triangles. We can divide the acute triangle and obtuse triangle into right angled triangles. A right triangle has three sides and three angles.

The angles are $\mathrm{\angle\:ABC\:,\:\angle\:BCA\:,\:\angle\:CAB}$ and the sides are AB, BC, CA

In a right angle triangle, Pythagoras theorem plays an important role to find the missing side measurement.

$\mathrm{AC^{2}\:=\:AB^{2}\:+\:BC^{2}}$

Where AC is the hypotenuse side.

The sum of three interior angles in a triangle measures 180° i.e., $\mathrm{\angle\:A\:+\:\angle\:B\:+\:\angle\:C\:=\:180°}$

Construction of right triangle

To construct a right angled triangle, we need measurement of two sides of a triangle. Among two sides, one must be hypotenuse and any one of the other two sides must be given. As we need to draw a right angled triangle at one angle, $\mathrm{\angle\:PQR\:is\:90°}$

When 2 sides are given

To construct a right angled triangle, when 2 sides are given other than hypotenuse, the steps are follows:

Draw a right angled triangle, the sides measure 8cm in the base and 6cm in the length.

Step 1 − Draw a line segment LM = 8cm

Step 2 − Take a compass, draw an arc from L as centre that cuts both the sides of the line

Step 3 − Draw two arcs that intersect each other.

Step 4 − Connect the lines LZ to draw a pendicular line at 90°

Step 5 − Measure 6cm in compass and draw an arc from L mark it as N.

Step 6 − Draw a line that connects the points N and M.

When base and hypotenuse given

To construct a right angled triangle from a base and hypotenuse, the steps are follows −

Given that PR = 7cm and QR = 5cm. Construct a ∆PQR, right angled at $\mathrm{\angle\:Q}$

Step 1 − Draw a line taking two points QR such that QR = 5cm

Step 2 − Take a compass, with any measurement draw two arcs on the both sides of the line from the point as centre Q.

Step 3 − Take a measurement more than half from Q and draw two arcs.

Step 4 − Connect the centre of the arc and point Q, draw a line QZ measures 90°.

Step 5 − Placing the compass measures 7cm at R,draw an arc in the line QZ

Step 6 − Mark the midpoint point P, now draw line PR.

When base and an angle on the base given

Construct a right angled triangle LMN whose base is 9cm and base angle is 60°.

Step 1 − Draw a line segment LM such that LM = 9cm

Step 2 − Draw a perpendicular line XLM= 90°

Step 3 − Draw an angle measuring 60° from $\mathrm{\angle\:ZRQ\:=\:60°}$ by drawing arcs using a compass.

Step 4 − Draw a line extended from 60° to connect the line segment LX to form MN.

A right angled triangle LMN is formed by a given base and base angle.

When the base and an angle not on the base given

Construct a right angled triangle use only ruler and compass, given a base AB = 5cm and angle opposite to base = 45°

$\mathrm{sum\:of\:three\:angles\:=\:180°}$

$\mathrm{45°\:+\:90°\:+\:x\:=\:180°}$

$\mathrm{x\:=\:45°}$

The another angle on the base is 45°

Step 1 − Draw a line segment measuring AB = 5cm.

Step 2 − Draw a perpendicular line from $\mathrm{\angle\:A\:=\:90°}$

Step 3 − Place the compass at B draw another perpendicular line.

Step 4 − To draw the angle $\mathrm{\angle\:B\:=\:45°}$ with the same measurements place the compass at arc on the line segment ZB draw an arc

Step 5 − Similarly draw another arc by placing the compass at arc on the line segment AB.

Step 6 − Extend the line to draw the line BC joins the line AX.

A right angled triangle ABC is formed.

Solved examples

1. Find the value of missing side

Solution

To find the unknown side, we have to apply Pythagoras theorem,

$\mathrm{AC^{2}\:=\:AB^{2}\:+\:BC^{2}}$

$\mathrm{(5)^{2}\:=\:x^{2}\:+\:(3)^{2}}$

$\mathrm{25\:=\:x^{2}\:+\:9}$

$\mathrm{x\:=\:4cm}$

2. Find the type of a triangle, if only two angle $\mathrm{\angle\:A\:=\:50°\:and\:\angle\:C\:=\:40°}$ given

Solution

To find the other side of a triangle

The sum of three interior angles in a triangle measures 180°

i.e., $\mathrm{\angle\:A\:+\:\angle\:B\:+\:\angle\:C\:=\:180°}$

$\mathrm{50\:+\:x\:+\:40\:=\:180}$

$\mathrm{x\:=\:90°}$

The triangle formed is a right angled triangle.

Conclusion

The right angled triangle has adjacent, opposite and hypotenuse sides. It has 3 angles, one among them must be 90°. The different ways of constructing a right angled triangle based on the sides and angles are possible using ruler and compass. Using the perpendicular bisector method of construction, we can draw 90°.

FAQs

1. What is the formula to find the area of a right angled triangle?

Area of a right angled triangle is same as the area of triangle

$\mathrm{Area\:of\:right\:angled\:Ltriangle\:=\:\frac{1}{2}(base\:\times\:height)}$

2. How to draw 45° in a triangle without using a protector?

Place the compass on a point and draw an arc named the arc. On the drawn arc, draw two arcs. Now draw two arcs that intersect each other. Draw a line to connect the points.

3. How to identify the sides in a right angled triangle?

The longer side in a right angled triangle which in the slanting line is hypotenuse. The other two angles change each time based on the angle taken. The side touching the angle is the adjacent side, the other is the opposite side.

4. Which side of a right angled triangle is opposite to 90°?

The hypotenuse is the longest side of a triangle which is opposite to the right angle.

5. Do we have an obtuse angle in a right angled triangle?

An obtuse angle measures more than 90°. A right angle has one 90° in it and the sum of the other two sides must be equal to 90°. A right angled triangle has 2 acute angles and one right angle in it.