Unit 11: Review Exercises

Review Exercise 11 Solutions

Comprehensive solved review problems combining multiple choice questions, triangle construction, and advanced locus shading problems.

Question 1 Standard MCQ Answers
Q# MCQ Statement Answer Key
i Sum of measures of any two sides of a triangle is ___ the measure of third side. greater than (b)
ii An equilateral triangle ___ can be isosceles (a)
iii If sum of measures of two angles is less than 90°, then the triangle is ___ obtuse angled (c)
iv The line segment joining the midpoint of a side to opposite vertex is called ___ median (a)
v The angle bisectors of a triangle intersect at ___ one point (a)
vi Locus of points equidistant from a fixed point is ___ circle (a)
vii Locus of points equidistant from two fixed points is ___ perpendicular bisector (b)
viii Locus of points equidistant from a fixed line is / are ___ parallel lines (d)
ix Locus of points equidistant from two intersecting lines is ___ angle bisector (c)
x Farther than 2 km from a fixed point B is region outside circle of radius ___ 2 km (c)
Questions 2 - 4 Geometry Constructions
Q.2 Construct a right angled triangle with measures of sides 6cm, 8cm and 10cm.
B A C 6cm 8cm 10cm
Steps of Construction
  • Draw line segment AB of length 6cm.
  • At vertex B, construct an angle of 90°.
  • With centre B and radius 8cm, draw an arc cutting the perpendicular ray at point C.
  • Join C to A. Segment AC will measure 10cm by Pythagoras' theorem.
Q.3 Construct a triangle ABC with mAB = 5.3 cm, m∠A = 30° and m∠B = 120°. Draw right bisector of AB.
A B C 5.3cm 30° 120°
Steps of Construction
  • Draw line segment AB of length 5.3cm.
  • Construct an angle of 30° at point A and draw a ray.
  • Construct an angle of 120° at point B and draw a ray. These rays intersect at point C.
  • Draw the perpendicular right bisector line of the base segment AB.
Q.4 Construct a triangle with mDE = 7.3cm, m∠D = 42° and mEF = 5.4cm.

This is an **ambiguous case (SSA)** because the side opposite the angle (EF = 5.4cm) is shorter than the adjacent side (DE = 7.3cm) but longer than the altitude from E (\(h = 7.3 \sin 42^\circ \approx 4.88\)cm). This yields two valid intersection points F and F'.

Steps of Construction
  • Draw segment DE of length 7.3cm.
  • At vertex D, draw an angle of 42°.
  • With centre E and radius 5.4cm, draw an arc cutting the angle ray at points F and F'.
  • Join E to F and E to F' to form the two valid triangles ΔDEF and ΔDEF'.
Questions 5 - 8 Triangles & Regions
Q.5 Construct a triangle XYZ with mYX = 8cm, mYZ = 7 cm and mXZ = 6.5 cm. Draw a locus of all points which are equidistant from XY and XZ.
Steps of Construction
  • Construct ΔXYZ using three sides SSS configuration.
  • The locus of points equidistant from lines XY and XZ is the **angle bisector of vertex X**.
  • Draw this bisector using compass arcs centered at vertex X.
Q.6 Construct triangle FGH with sides 6.4cm, angle 122°. Draw bisector loci.
Steps of Construction
  • Draw base segment FG = 6.4cm. Construct angle 122° at G, cut at GH = 6.4cm to form isosceles triangle ΔFGH.
  • Draw right bisector of FG (points equidistant from F and G).
  • Draw angle bisector of ∠G (points equidistant from arms FG and GH).
  • Mark their intersection as point O.
Q.7 Two houses Q and R are 73 metres apart. Locus 32m from Q and 48m from the line QR.
Geometric Analysis:
The locus 32m from Q is a circle of radius 3.2cm. The locus 48m from line QR consists of parallel lines at a distance of 4.8cm from QR. Since the circle has a maximum reach of 3.2cm from QR, it lies completely within the band between the two parallel lines. Thus, the loci do not intersect, and there are **no points** that satisfy both conditions simultaneously.
Q.8 Rectangle field ABCD (\(70\text{m} \times 60\text{m}\)). Show the region inside the field which is less than 30m from C and farther than 25m from AB.
A B C D Valid Region
Steps of Construction
  • Using scale 1cm = 10m, draw rectangle ABCD of dimensions 7cm × 6cm.
  • With centre C, draw a quarter circle of radius 3cm. The interior of this arc is the region less than 30m from C.
  • Draw line segment PQ parallel to AB at a distance of 2.5cm. The area above this line is farther than 25m from AB.
  • The intersection of these two regions (the shaded quarter-circle sector lying above the line PQ) is the required region.
Interactive Review Sandbox Assess & Explore
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