In a quadrilateral \( \mathrm{ABCD}, \mathrm{AB}=4.7 \mathrm{~cm}, \mathrm{BC}=5.3 \mathrm{~cm}, \mathrm{CD}=6.1 \mathrm{~cm}, \mathrm{AD}=5.9 \mathrm{~cm} \) and \( \mathrm{BD}=4.4 \mathrm{~A} \) (i) Construct a quadrilateral ABCD from the given data: (ii) Measure the length of AC . In a quadrilateral \( \mathrm{DEFG}, \mathrm{DE}=3.5 \mathrm{~cm}, \mathrm{EF}=6.5 \mathrm{~cm}, \Varangle \mathrm{D}=\Varangle \mathrm{F}=105^{\circ} \) and \( \not \subset \mathrm{G}=75^{\circ} \), (i) Construct a quadrilateral DEFG from the given data: (ii) Measure the unknown sides In a quadrilateral \( \mathrm{DEFG}, \mathrm{DE}=5.6 \mathrm{~cm}, \mathrm{EF}=4 \mathrm{~cm}, \Varangle \mathrm{D}=40^{\circ}, \Varangle \mathrm{E}=110^{\circ} \), and \( \not \boxed{\mathrm{G}}=70^{\circ} \) (i) Construct a quadrilateral DEFG from the given data: (ii) Measure the unknown sides

To solve the problem, we will break it down into three parts, each corresponding to the quadrilaterals \(ABCD\) and \(DEFG\). ### Quadrilateral ABCD **Given:** - \(AB = 4.7 \, \text{cm}\) - \(BC = 5.3 \, \text{cm}\) - \(CD = 6.1 \, \text{cm}\) - \(AD = 5.9 \, \text{cm}\) - \(BD = 4.4 \, \text{cm}\) #### (i) Constructing Quadrilateral ABCD 1. **Draw Line Segment AB:** - Draw a line segment \(AB\) of length \(4.7 \, \text{cm}\). 2. **Construct Point B:** - From point \(A\), use a compass to measure \(5.9 \, \text{cm}\) (length of \(AD\)) and draw an arc to locate point \(D\). 3. **Construct Point C:** - From point \(B\), measure \(5.3 \, \text{cm}\) (length of \(BC\)) and draw an arc. - From point \(D\), measure \(6.1 \, \text{cm}\) (length of \(CD\)) and draw another arc. - The intersection of these two arcs gives point \(C\). 4. **Connect Points:** - Connect points \(A\), \(B\), \(C\), and \(D\) to form quadrilateral \(ABCD\). #### (ii) Measuring Length of AC To measure the length of \(AC\), use a ruler to measure the distance between points \(A\) and \(C\) after constructing the quadrilateral. ### Quadrilateral DEFG (First Instance) **Given:** - \(DE = 3.5 \, \text{cm}\) - \(EF = 6.5 \, \text{cm}\) - \(\angle D = \angle F = 105^\circ\) - \(\angle G = 75^\circ\) #### (i) Constructing Quadrilateral DEFG 1. **Draw Line Segment DE:** - Draw a line segment \(DE\) of length \(3.5 \, \text{cm}\). 2. **Construct Angles:** - At point \(D\), use a protractor to measure and draw an angle of \(105^\circ\) to locate point \(F\). - At point \(E\), measure and draw another angle of \(105^\circ\) to locate point \(G\). 3. **Construct Lengths:** - From point \(D\), measure \(6.5 \, \text{cm}\) to locate point \(F\). - From point \(E\), measure the distance to locate point \(G\) based on the angle. 4. **Connect Points:** - Connect points \(D\), \(E\), \(F\), and \(G\) to form quadrilateral \(DEFG\). #### (ii) Measuring Unknown Sides After constructing the quadrilateral, measure the lengths of the unknown sides \(DG\) and \(EG\) using a ruler. ### Quadrilateral DEFG (Second Instance) **Given:** - \(DE = 5.6 \, \text{cm}\) - \(EF = 4 \, \text{cm}\) - \(\angle D = 40^\circ\) - \(\angle E = 110^\circ\) - \(\angle G = 70^\circ\) #### (i) Constructing Quadrilateral DEFG 1. **Draw Line Segment DE:** - Draw a line segment \(DE\) of length \(5.6 \, \text{cm}\). 2. **Construct Angles:** - At point \(D\), use a protractor to measure and draw an angle of \(40^\circ\) to locate point \(F\). - At point \(E\), measure and draw an angle of \(110^\circ\) to locate point \(G\). 3. **Construct Lengths:** - From point \(D\), measure \(4 \, \text{cm}\) to locate point \(F\). - From point \(E\), measure the distance to locate point \(G\) based on the angle. 4. **Connect Points:** - Connect points \(D\), \(E\), \(F\), and \(G\) to form quadrilateral \(DEFG\). #### (ii) Measuring Unknown Sides After constructing the quadrilateral, measure the lengths of the unknown sides \(DG\) and \(EG\) using a ruler. ### Summary - For each quadrilateral, the construction involves drawing the specified lengths and angles accurately. - The measurement of unknown sides can be done using a ruler after the construction is complete. If you need further assistance with specific calculations or measurements, please let me know!