Using a ruler and a pair of compasses only construct i) a quadrilateral \( A B C D,|A B|=8 \mathrm{~cm} \), \( |A D|=6 \mathrm{~cm},|B C|=10 \mathrm{~cm} \), \( \angle B A D=60^{\circ} \) and \( \angle A D C=135^{\circ} \). ii) the locus \( l_{1} \) of points equidistant from \( B C \) and \( C D \). iii) The line, \( l_{2} \), from \( B \) perpendicular to \( l_{1} \). b) i) Locate \( E \), the point of intersection of

Constructing geometric figures with precise measurements and angles using only a ruler and compass requires careful step-by-step procedures. Below are the detailed instructions for each part of your problem. --- ## **a) i) Constructing Quadrilateral \( ABCD \)** **Given:** - \( |AB| = 8 \) cm - \( |AD| = 6 \) cm - \( |BC| = 10 \) cm - \( \angle BAD = 60^\circ \) - \( \angle ADC = 135^\circ \) ### **Step-by-Step Construction:** 1. **Draw Base \( AB \):** - Use the ruler to draw a straight horizontal line segment \( AB \) of length 8 cm. 2. **Construct \( \angle BAD = 60^\circ \):** - At point \( A \), use a protractor to construct a \( 60^\circ \) angle above the base \( AB \). - Alternatively, using a compass: - With \( A \) as the center, draw an arc intersecting \( AB \) at some point \( F \). - Without changing the compass width, place the compass at \( F \) and draw another arc intersecting the first arc at point \( G \). - Draw line \( AG \); this line makes a \( 60^\circ \) angle with \( AB \). 3. **Draw Side \( AD \):** - Set the compass to 6 cm. - From point \( A \), draw an arc intersecting the \( 60^\circ \) line at point \( D \). - Connect \( A \) to \( D \) to form side \( AD \). 4. **Construct \( \angle ADC = 135^\circ \):** - At point \( D \), construct a \( 135^\circ \) angle with side \( AD \). - Using a protractor, measure \( 135^\circ \) from \( AD \) towards the interior where point \( C \) will lie. - Alternatively, using compass: - Since \( 135^\circ = 90^\circ + 45^\circ \), first construct a perpendicular to \( AD \) at \( D \). - Then, bisect the right angle to get a \( 45^\circ \) angle, resulting in a \( 135^\circ \) angle from \( AD \). 5. **Draw Side \( DC \):** - From point \( D \), draw a line along the \( 135^\circ \) angle. - The length of \( DC \) will be determined based on the intersection with the next step. 6. **Construct Side \( BC = 10 \) cm:** - From point \( B \), use the compass set to 10 cm to draw an arc. - From point \( C \), the intersection of \( DC \) and the 10 cm arc from \( B \) determines the position of \( C \). - Alternatively, adjust the compass to ensure \( BC = 10 \) cm intersects appropriately. 7. **Connect Points to Form Quadrilateral \( ABCD \):** - Once points \( A \), \( B \), \( C \), and \( D \) are determined, connect them in order to form quadrilateral \( ABCD \). ### **Result:** You will have a quadrilateral \( ABCD \) with the specified side lengths and angles. --- ## **a) ii) Constructing the Locus \( l_{1} \) of Points Equidistant from \( BC \) and \( CD \)** **Definition:** The locus of points equidistant from two lines is the angle bisector of the angle formed by those lines. ### **Step-by-Step Construction:** 1. **Identify Lines \( BC \) and \( CD \):** - Ensure that lines \( BC \) and \( CD \) are clearly drawn and extended if necessary. 2. **Construct the Angle Bisector:** - At the intersection point \( C \) of \( BC \) and \( CD \), construct the angle bisector of \( \angle BCD \). - **Using Compass and Straightedge:** 1. Place the compass at point \( C \) and draw an arc that intersects both \( BC \) and \( CD \) at points \( E \) and \( F \), respectively. 2. Without changing the compass width, draw arcs from points \( E \) and \( F \) that intersect each other at point \( G \). 3. Draw a straight line from \( C \) through point \( G \); this is the angle bisector \( l_{1} \). ### **Result:** The line \( l_{1} \) is the angle bisector of \( \angle BCD \), representing all points equidistant from \( BC \) and \( CD \). --- ## **a) iii) Constructing the Line \( l_{2} \) from \( B \) Perpendicular to \( l_{1} \)** **Objective:** To draw a line from point \( B \) that is perpendicular to the previously constructed locus \( l_{1} \). ### **Step-by-Step Construction:** 1. **Identify Line \( l_{1} \):** - Ensure that the angle bisector \( l_{1} \) is accurately drawn. 2. **Construct Perpendicular at \( B \):** - Use the compass and straightedge method to erect a perpendicular line at point \( B \) relative to line \( l_{1} \). - **Using Compass and Straightedge:** 1. Place the compass at point \( B \) and draw an arc that intersects \( l_{1} \) at two points, say \( P \) and \( Q \). 2. With the same compass width, draw arcs from points \( P \) and \( Q \) above or below \( l_{1} \) to create points \( R \) and \( S \). 3. Draw line \( l_{2} \) from \( B \) through the point where the two arcs intersect (either \( R \) or \( S \)). 4. This line \( l_{2} \) is perpendicular to \( l_{1} \). ### **Result:** Line \( l_{2} \) is the perpendicular line from point \( B \) to the locus \( l_{1} \). --- ## **b) i) Locating Point \( E \), the Point of Intersection of …** *It appears that the description for part **b) i)** was incomplete. To provide a precise construction or explanation, additional information about what lines or elements intersect at point \( E \) is needed.* --- **Note:** For accurate constructions, ensure that each step is followed meticulously, and measurements are precise. Using a sharp pencil and maintaining steady handwork will aid in achieving the desired geometric figures.