KCSE 2025 Maths P1 Q10 — Construct Triangle ABC and Measure Angle ABC

KCSE 2025 Form 4 Geometry

Published

The Question

“Line AB equals 5 cm is one side of triangle ABC in which angle BAC is 120 degrees and line BC is 10 cm. Using a pair of compasses and a ruler, complete triangle ABC and hence measure the size of angle ABC.”

1

Draw the base AB

Start with the side you know exactly. Use a ruler to draw a straight line segment AB that is 5 centimetres long, and label the two ends A and B. Everything else is built onto this base, so make it accurate.

AB=5 cmAB = 5\ \text{cm}
2

Construct the 120 degree angle at A

The angle BAC is given, so build it at A. Using a protractor place a mark at 120 degrees from AB, then draw a long ray starting at A through that mark. Point C must lie somewhere along this ray, so make the ray long enough to catch it.

∠BAC=120∘\angle BAC = 120^\circ
3

Locate C with an arc from B

The only condition left is that C is 10 cm from B. Open your compasses to a radius of 10 centimetres, place the sharp point on B, and swing an arc that cuts the ray you drew from A. The point where the arc crosses the ray is C, because it is both on the 120 degree ray and exactly 10 cm from B.

BC=10 cmBC = 10\ \text{cm}
4

Complete the triangle and measure angle ABC

Join B to C with a straight line to close the triangle. Then place a protractor at B and read the angle between BA and BC. The measured value is about 34 degrees.

∠ABC≈34∘\angle ABC \approx 34^\circ
5

Confirm by the sine rule

You can check the measurement by calculation. Side BC is opposite angle A and side AB is opposite angle C, so the sine rule links them. Rearranging gives the sine of angle C, then angle C itself, and finally angle B from the fact that the three angles of a triangle add to 180 degrees.

sin⁡CAB=sin⁡ABC\frac{\sin C}{AB} = \frac{\sin A}{BC}
sin⁡C=5×sin⁡120∘10=0.433\sin C = \frac{5 \times \sin 120^\circ}{10} = 0.433
C≈25.7∘C \approx 25.7^\circ
∠ABC=180∘−120∘−25.7∘=34.3∘\angle ABC = 180^\circ - 120^\circ - 25.7^\circ = 34.3^\circ

The calculated 34.3 degrees agrees with the protractor reading of about 34 degrees.

Final Result

Triangle ABC is completed by joining C, and the measured size of angle ABC is about 34 degrees (34.3 degrees by calculation).

Why this method works

The construction works because a triangle is fully fixed once two sides and the angle between the known lines are pinned down. The 5 cm base and the 120 degree ray fix the direction of side AC, while the 10 cm arc from B fixes how far along that direction C must sit. Their intersection is the only point satisfying both conditions, so the triangle is unique. The sine rule then verifies the drawing independently: it relates each side to the sine of the angle opposite it, letting you compute angle C and, using the 180 degree angle sum, angle ABC without measuring.

✓

Angles add up: 120 + 25.7 + 34.3 = 180 degrees, confirming the result.