Two line segments \(AB\) and \(AC\) include an angle of \(60^\circ\) where \(AB=5\,\text{cm}\) and \(AC=7\,\text{cm}\). Locate points \(P\) on \(AB\) and \(Q\) on \(AC\) such that \(AP=\dfrac{3}{4}AB\) and \(AQ=\dfrac{1}{4}AC\). Join \(P\) and \(Q\) and find \(PQ\).
Final answer: \(PQ=\dfrac{13}{4}\,\text{cm}=3.25\,\text{cm}.\)
Draw a parallelogram \(ABCD\) in which \(BC=5\,\text{cm}\), \(AB=3\,\text{cm}\) and \(\angle ABC=60^\circ\). Divide it into triangles \(BCD\) and \(ABD\) by diagonal \(BD\). Construct \(\triangle BD'C'\) similar to \(\triangle BDC\) with scale factor \(\dfrac{4}{3}\). Draw \(D'A'\parallel DA\) with \(A'\) on the extension of \(BA\). Decide whether \(A'BC'D'\) is a parallelogram.
Final answer: Yes, \(A'BC'D'\) is a parallelogram.
Draw two concentric circles of radii \(3\,\text{cm}\) and \(5\,\text{cm}\). From a point \(T\) on the outer circle, construct the pair of tangents to the inner circle. Measure the length of a tangent and verify it by calculation.
Final answer: Each tangent has length \(4\,\text{cm}.\)
Draw an isosceles triangle \(ABC\) with \(AB=AC=6\,\text{cm}\) and \(BC=5\,\text{cm}\). Construct \(\triangle PQR\sim \triangle ABC\) such that \(PQ=8\,\text{cm}\). Also justify the construction.
Final answer: Scale factor \(k=\dfrac{8}{6}=\dfrac{4}{3}\). Hence \(PR=8\,\text{cm}\) and \(QR=\dfrac{4}{3}\times 5=\dfrac{20}{3}\,\text{cm}\approx6.67\,\text{cm}.\)
Draw a triangle \(ABC\) with \(AB=5\,\text{cm}\), \(BC=6\,\text{cm}\) and \(\angle ABC=60^\circ\). Construct a triangle similar to \(\triangle ABC\) with scale factor \(\dfrac{5}{7}\). Justify the construction.
Final answer: The required triangle is a reduction of \(\triangle ABC\) by factor \(\dfrac{5}{7}\); each side equals \(\dfrac{5}{7}\) of the corresponding side of \(\triangle ABC\).
Draw a circle of radius \(4\,\text{cm}\). Construct a pair of tangents to it such that the angle between the tangents is \(60^\circ\). Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of the tangents.
Final answer: Distance from the centre to the intersection point is \(OT=2r=8\,\text{cm}.\)
Draw \(\triangle ABC\) in which \(AB=4\,\text{cm}\), \(BC=6\,\text{cm}\) and \(AC=9\,\text{cm}\). Construct a triangle similar to \(\triangle ABC\) with scale factor \(\dfrac{3}{2}\). Justify the construction. Are the two triangles congruent?
Final answer: The enlarged triangle has sides \(AB'=6\,\text{cm}\), \(BC'=9\,\text{cm}\), \(AC'=13.5\,\text{cm}\). They are not congruent (scale factor \(\neq 1\)).