**Revision Notes on Practical Geometry**

**Constructing a Quadrilateral**

A quadrilateral has some measurements like - 4 sides, 4 angles and 2 diagonals.

We can construct a unique quadrilateral if we know the five measurements.

**1. If the four sides and a diagonal of the quadrilateral are given.**

**Example**

Construct a quadrilateral ABCD in which AB = 5 cm, BC = 7 cm, CD = 6 cm, DA = 6.5 cm and AC = 8 cm.

**Solution**

**Step 1:** ∆ABC can be constructed using SSS criterion of the construction of triangle.

**Step 2:** Here we can see that AC is diagonal, so D will be somewhere opposite to B with reference to AC.

AD = 6.5 cm so draw an arc from A as the centre with radius 6.5 cm.

**Step 3:** Now draw an arc with C as the centre and by taking radius 6 cm so that it intersects the above arc.

**Step 4:** The point of intersection of the two arcs is point D. Now join AD and DC to complete the quadrilateral.

Hence, ABCD is the required quadrilateral.

**2. If two diagonals and three sides of the quadrilateral are given**

**Example**

Construct a quadrilateral ABCD if the two diagonals are AC = 6.5 cm and BD = 8 cm. The other sides are BC = 5.5 cm, AD = 6.5 cm and CD = 6 cm.

**Solution**

First of all, draw a rough sketch of the quadrilateral by using the given measurements. Then start constructing the real one.

**Step 1:** We can see that AD, AC and DC are given so we can construct a triangle ΔACD by using SSS criterion.

**Step 2:** Now, we know that BD is given so we can draw the point B keeping D as the centre and draw an arc of radius 8 cm just opposite to the point D with reference to AC.

**Step 3:** BC is given so we can draw an arc keeping C as centre and radius 5.5 cm so that it intersects the other arc.

**Step 4:** That point of intersection of the arcs is point B. Join AB and BC to complete the quadrilateral.

ABCD is the required quadrilateral.

**3.** **If three angles and two adjacent sides of the quadrilateral are given.**

**Example**

Construct a quadrilateral ABCD in which the two adjacent sides are AB = 4.5 cm and BC = 7.5 cm. The given three angles are ∠A = 75ᵒ, ∠B = 105ᵒ and ∠C = 120ᵒ.

**Solution**

Draw a rough sketch so that we can construct easily.

**Step 1: **Draw AB = 4.5 cm. Then measure ∠B = 105° using protractor and draw BC = 7.5 cm.

**Step 2:** Draw ∠C = 120°.

**Step 3:** Measure ∠A = 75° and make a line until it touches the line coming from point C.

ABCD is the required quadrilateral.

**4. If the three sides with two included angles of the quadrilateral are given.**

**Example**

Construct a quadrilateral ABCD in which the three sides are AB = 5 cm, BC = 6 cm and CD = 7.5 cm. The two included angles are ∠B = 105° and ∠C = 80°.

**Solution**

Draw a rough sketch.

**Step 1:** Draw the line BC = 6 cm. Then draw ∠B = 105° and mark the length of AB = 5 cm.

**Step 2:** Draw ∠C = 80° using protractor towards point B.

**Step 3:** Mark the length of CD i.e.7.5 cm from C to make CD = 7.5 cm.

**Step 4:** Join AD which will complete the quadrilateral ABCD.

Hence ABCD is the required quadrilateral.

**Some Special Cases**

There are some special cases in which we can construct the quadrilateral with less number of measurements also.

**Example**

Construct a square READ with RE = 5.1 cm.

**Solution**

Given Re = 5.1 cm.

As it is a special quadrilateral called square, we can get more details out of it.

a. All sides of square are equal, so RE = EA = AD = RD = 5.1 cm.

b. All the angles of a square are 90°, so ∠R = ∠E = ∠A = ∠D = 90°

**Step 1**: Draw a rough sketch of the square.

**Step 2:** To construct a square, draw a line segment RE = 5.1 cm. Then draw the angle of 90° at both ends R and E of the line segment RE.

**Step 3:** As all the sides of the square READ are equal, draw the arc of 5.1 cm from the vertex R and E to cut the lines RD and EA respectively.

**Step 4:** Join A and D to make a line segment AD.

READ is the required square.