Question

Below you are given the description of two quadrilaterals. On a piece of paper (labelled and in order), use the appropriate tools (straightedge, compass, etc.) to construct the described quadrilaterals and the perpendicular bisectors of each of their sides. After each construction, state why there is no circle that will circumscribe the quadrilateral. 1. a parallelogram that is not a rectangle 2. a non-isoceles trapezoid. There is no diagram.

Answer 1

Construct a quadrilateral ABCD in which AB = 3.6 cm, ∠ABC = 80°, BC = 4 cm, ∠BAD = 120° and AD = 5 cm.

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Solution:

First we draw a rough sketch of quadrilateral ABCD and write down its dimensions, as shown (Rough Sketch) →

Steps of Construction:

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Step 1: Draw AB = 3.6 cm.

Step 2: Make ∠ABX = 80°.

Step 3: With B as center and radius equal to 4 cm, draw an arc, cutting BX at C.

Step 4: Make ∠BAY = 120°.

Step 5: With A as center and 5 cm as radius, draw an arc, cutting AY at D.Step 6: Join CD.

Then, ABCD is the required quadrilateral.

here a circle cannot be formed because the opposite angles cannot be supplementary(180⁰ degrees).

PERPENDICULAR BISECTOR:

the perpendicular bisectors can be drawn as follows

Take each vertex as center and make a radius of more than the corresponding side of the edge and draw an arc on either side of the edge,

now keeping the same radius taking the other vertex as center and draw an arc cutting the previous arc

now join the two intersecting points through the edge which is the required perpendicular bisector.

TRAPEZIUM:

Properties of a Trapezium

• The bases of the trapezium are parallel to each other (MN ⫽ OP).
• No sides, angles and diagonals are congruent.

Let's see the rough figure of the trapezium.

Here,

• AB and CD are parallel.
• The height of the trapezium is AD = 3.3cm.
• BC= 3.6 cm

Now the solution part:

• Draw a line AB=4.5 cm.

• Next using the compass construct an angle BAD=90°. Since, the distance between the parallel sides is 3.3cm, it implies that the line perpendicular to AB at A is 3.3 cm. So, cut an arc of 3.3 cm on the perpendicular line and name it D.

• We know that the two sides (AB and CD) are parallel. Therefore, construct an angle ADX=90°. Draw a line DX which is parallel to AB.

• Now we have three sides of the trapezium. So, cut an arc of 3.6 cm from B on line DX. Join B and C.

Thus, a trapezium with sides AB=4.5 cm, AD=3.3 cm, BC=3.6 cm and Angle B is obtuse.

As here, name of the trapezium not mentioned, I've supposed it as trap ABCD..

GIVEN: AB = 4.5 cm, BC= 3.6 cm, angle ABC is obtuse & height of the trapezium( ie, distance between the parallel sides) = 3.3cm…

Here, unless it's given either of the lengths of AD, BD or CD, or the measure of angle DAB,we are not going to get a unique trapezium.

As , i've half constructed a figure for you.., in which vertices A,B, & C can be located as per the requirements. AND the LOCUS of point D , will be parallel line L2 L4 . You may take the vertex D any where on this line.. BUT with following conditions:

(1) if you consider angle DAB, an acute angle, Then 3.3 < AD < 6.1 ( This can be calculated by calculating the length of AC)

(2) if you consider angle DAB, a right angle, then AD = 3.3 cm

(3) And if you consider angle DAB, an obtuse angle, then D lies on L2 L4, in such a way that angle DAL3 > 0° & less than 90°

Here also a circle can not be drawn as the oppposite angles are not supplementary and the a pair non-parallel sides are not equal.