Unit 11: Practical Geometry - Triangles

Exercise 11.1 Solutions

Clean, step-by-step solved exercises formatted with beautiful styling, custom vector graphics, and standard mathematical proofs.

Question 1 Perpendicular Bisectors
Construct ΔABC with the given measurements and verify that the perpendicular bisectors of the triangle are concurrent.
Part (i) Acute Triangle
Given: mAB = 5cm, mBC = 6cm, mAC = 7cm.
O A B C 5cm 6cm 7cm
Steps of Construction
  • Draw a line segment AB of length 5cm.
  • With centre A and radius 7cm, draw an arc above AB.
  • With centre B and radius 6cm, draw another arc intersecting the first arc at point C.
  • Join C to A and B to complete ΔABC.
  • Draw perpendicular bisectors of AB, BC, and AC.
  • We observe that all three perpendicular bisectors meet at a single point O inside the triangle.
Part (ii) Obtuse Triangle
Given: mAB = 7.1cm, m∠B = 135°, mBC = 6.5cm.
O A B C 7.1cm 6.5cm 135°
Steps of Construction
  • Draw a line segment AB of length 7.1cm.
  • At vertex B, construct an angle of 135° using a protractor/compass.
  • With centre B and radius 6.5cm, cut the angle ray at point C.
  • Join C to A to complete the obtuse triangle ΔABC.
  • Draw the perpendicular bisectors of the sides AB, BC, and AC.
  • We observe that the three perpendicular bisectors are concurrent at a point O located outside the triangle.
Question 2 Medians & Centroid
Construct ΔLMN of the following measurements and verify that the medians of the triangle are concurrent.
Part (i) ASA Method
Given: mLM = 4.9cm, m∠L = 51°, m∠M = 38°.
G L M N 51° 38° 4.9cm
Steps of Construction
  • Draw a line segment LM of length 4.9cm.
  • At vertex L, construct an angle of 51°.
  • At vertex M, construct an angle of 38°.
  • The rays of these two angles intersect at point N to complete ΔLMN.
  • Draw perpendicular bisectors of LM, MN, and LN to find their midpoints X, Y, and Z respectively.
  • Join vertices to their opposite midpoints: L to Y, M to Z, and N to X.
  • We observe that these three medians are concurrent at centroid G inside the triangle.
Part (ii) SSA Configuration
Given: mMN = 4.8cm, m∠N = 30°, mLM = 8.1cm.
G L M N 30° 4.8cm 8.1cm
Steps of Construction
  • Draw a line segment MN of length 4.8cm.
  • At vertex N, construct an angle of 30° and draw a ray.
  • With centre M and radius 8.1cm, draw an arc cutting the angle ray at point L.
  • Join L to M and N to complete the triangle ΔLMN.
  • Draw perpendicular bisectors of the sides to find the midpoints X (on LM), Y (on MN), and Z (on LN).
  • Draw the medians LY, MZ, and NX.
  • Observe that the medians intersect at centroid G, verifying concurrency.
Question 3 Angle Bisectors
Verify that the angle bisectors of ΔABC are concurrent with the following measurements.
Part (i) Incenter Exists
Given: mAB = 4.5cm, m∠A = 45°, mAC = 5.3cm.
I A B C 4.5cm 5.3cm 45°
Steps of Construction
  • Draw a line segment AB of length 4.5cm.
  • At vertex A, construct an angle of 45° using compass/protractor.
  • With centre A and radius 5.3cm, cut the angle ray at point C.
  • Join C to B to complete the triangle ΔABC.
  • Using a compass, draw the angle bisectors →AL, →BM, and →CN for ∠A, ∠B, and ∠C respectively.
  • We observe that these bisectors meet at incenter I, verifying concurrency.
Part (ii) Construction Impossible
Given: mAB = 6cm, m∠A = 150°, m∠B = 60°.
A B 6cm 150° 60° Rays Diverge!
Geometric Proof of Impossibility:
The sum of two angles in a triangle must be strictly less than 180°:
m∠A + m∠B = 150° + 60° = 210° > 180° Since the sum of these two angles exceeds the total angle sum allowed for any Euclidean triangle, the rays drawn from vertices A and B will diverge and never intersect. Hence, the construction of this triangle is impossible.
Question 4 Altitudes & Orthocenter
Given the measurements of ΔDEF: mDE = 4.8cm, mEF = 4cm and m∠E = 45°, draw altitudes of ΔDEF and find the orthocenter.
Solution Steps:
O D E F 4.8cm 4cm 45° R P Q
Steps of Construction
  • Draw line segment DE of length 4.8cm.
  • At vertex E, construct an angle of 45° using compass/protractor.
  • With centre E and radius 4.0cm, draw an arc cutting the angle ray at point F.
  • Join F to D to complete the triangle ΔDEF.
  • Draw altitude FR from vertex F perpendicular to the opposite side DE.
  • Draw altitude DP from vertex D perpendicular to the opposite side EF.
  • Draw altitude EQ from vertex E perpendicular to the opposite side DF.
  • The point of intersection of the three altitudes is the orthocenter O, located inside the triangle.
Question 5 Ambiguous Case (SSA)
Construct the following triangles and find whether there exists any ambiguity.
Part (i) 2 Triangles Exist
Given: ΔBCD, mBC = 5cm, m∠B = 62°, mCD = 4.7cm.
B C D D' 5cm 4.7cm 4.7cm 62°
Steps of Construction
  • Draw line segment BC of length 5cm.
  • At vertex B, construct an angle of 62° using a protractor.
  • With centre C and radius 4.7cm, draw an arc intersecting the angle ray at two points D and D'.
  • Join C to D and C to D'.
  • Thus, two distinct triangles (ΔBCD and ΔBCD') can be constructed. This confirms that this is an ambiguous case.
Part (ii) 2 Triangles Exist
Given: ΔLMN, mLM = 6cm, m∠M = 42°, mLN = 5cm.
M L N N' 6cm 5cm 5cm 42°
Steps of Construction
  • Draw line segment ML of length 6cm.
  • At vertex M, construct an angle of 42° and draw a ray.
  • With centre L and radius 5cm, draw an arc intersecting the angle ray at two points N and N'.
  • Join L to N and L to N'.
  • Thus, two distinct triangles (ΔLMN and ΔLMN') can be constructed. This confirms that this is an ambiguous case.
Interactive Sandbox Explore Triangle Centers
Adjust the base length, angles, and sides of the triangle below. Toggle the centers and lines (perpendicular bisectors, medians, angle bisectors, altitudes) to see how they behave in real-time. Try creating an obtuse angle to watch the orthocenter and circumcenter jump outside the triangle!
Overlay Construction Lines:

Live Geometry Report

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