To divide a line segment AB in the ratio 5:7, first a ray AX is drawn so that ∠BAX is an acute angle and then at equal distances points are marked on the ray AX such that the minimum number of these points is
8
10
11
12
To divide a line segment AB in the ratio 4:7, a ray AX is drawn first such that ∠BAX is an acute angle and then points A₁, A₂, A₃, ... are located at equal distances on the ray AX and the point B is joined to
A12
A11
A10
A9
To divide a line segment AB in the ratio 5:6, draw a ray AX such that ∠BAX is an acute angle, then draw a ray BY parallel to AX and the points A₁, A₂, A₃, ... and B₁, B₂, B₃, ... are located at equal distances on rays AX and BY, respectively. Then the points joined are
A5 and B6
A6 and B5
A4 and B5
A5 and B4
To construct a triangle similar to a given ΔABC with its sides 3⁄7 of the corresponding sides of ΔABC, first draw a ray BX such that ∠CBX is an acute angle and X lies on the opposite side of A with respect to BC. Then locate points B₁, B₂, B₃, ... on BX at equal distances and next step is to join
B10 to C
B3 to C
B7 to C
B4 to C
To construct a triangle similar to a given ΔABC with its sides 8⁄5 of the corresponding sides of ΔABC, draw a ray BX such that ∠CBX is an acute angle and X is on the opposite side of A with respect to BC. The minimum number of points to be located at equal distances on ray BX is
5
8
13
3
To draw a pair of tangents to a circle which are inclined to each other at an angle of 60°, it is required to draw tangents at end points of those two radii of the circle, the angle between them should be
135°
90°
60°
120°
By geometrical construction, it is possible to divide a line segment in the ratio \(\sqrt{3} : \dfrac{1}{\sqrt{3}}\).
To construct a triangle similar to a given \(\triangle ABC\) with its sides \(\dfrac{7}{3}\) of the corresponding sides of \(\triangle ABC\), draw a ray \(BX\) making an acute angle with \(BC\) and with \(X\) on the opposite side of \(A\) w.r.t. \(BC\). Locate points \(B_1,B_2,\dots,B_7\) at equal distances on \(BX\). Then join \(B_3\) to \(C\) and draw \(B_6C'\) \(\parallel\) \(B_3C\) meeting \(BC\) produced at \(C'\). Finally, draw \(A'C'\) \(\parallel\) \(AC\).
A pair of tangents can be constructed from a point \(P\) to a circle of radius \(3.5\,\text{cm}\) situated at a distance of \(3\,\text{cm}\) from the centre.
A pair of tangents can be constructed to a circle inclined at an angle of \(170^\circ\).
Draw a line segment of length 7 cm. Find a point \(P\) on it which divides it in the ratio \(3:5\).
Check (optional): \(AP = \dfrac{3}{8}\times 7 = 2.625\,\text{cm},\; BP = \dfrac{5}{8}\times 7 = 4.375\,\text{cm}.\)
Draw a right triangle \(ABC\) in which \(BC=12\,\text{cm},\; AB=5\,\text{cm}\) and \(\angle B=90^{\circ}\). Construct a triangle similar to it with scale factor \(\dfrac{2}{3}\). Is the new triangle also a right triangle?
Yes. Similarity preserves angles, hence the image of \(\angle B=90^{\circ}\) is also \(90^{\circ}\); the new triangle is right-angled at \(B\).
Draw a triangle \(ABC\) in which \(BC=6\,\text{cm},\; CA=5\,\text{cm}\) and \(AB=4\,\text{cm}.\) Construct a triangle similar to it with scale factor \(\dfrac{5}{3}\).
Construct a tangent to a circle of radius \(4\,\text{cm}\) from a point which is at a distance of \(6\,\text{cm}\) from its centre.
Length (by right triangle \(\triangle OPT\)): \[ PT=\sqrt{OP^2-OT^2}=\sqrt{6^2-4^2}=\sqrt{20}=2\sqrt{5}\,\text{cm}. \]
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 \(\neq 1\)).