SPM 2024 Add Maths Paper 2, Question 12Solution of Triangles

Solved on video in English · Form 4, Solution of Triangles · 9:28 video

The question

(a)

Diagram 6.1 shows the triangular cardboard MFH.

(a) Based on Diagram 6.1, derive the cosine rule for side MH.

Based on Diagram 6.1, derive the cosine rule for side MH.

(b)(i)

(b) The cardboard is placed into a cuboid as shown in Diagram 6.2. It is given that M is the midpoint of side AB.

(i) Find ∠MFH.

Find ∠MFH.

(b)(ii)(a)

(ii) It is given that △M′F′H′ has a different shape from △MFH such that M′F′ = MF, M′H′ = MH and ∠M′H′F′ = ∠MHF.

(a) Sketch △M′F′H′.

Sketch △M′F′H′.

(b)(ii)(b)

(b) Find ∠F′M′H′.

Hence, calculate the area, in cm², of △M′F′H′.

Find ∠F′M′H′.

Hence, calculate the area, in cm², of △M′F′H′.

Part (a)

Step 1

Drop a perpendicular from M to line HF; let the foot be X, with FX = x, so HX = m − x.

Why this step

A general triangle has no right angle. Dropping a perpendicular makes two right triangles, so Pythagoras and cos suddenly work.

7 more steps, each with the reason for it, in the full solution

Final answer

Derived: MH² = MF² + HF² − 2(MF)(HF) cos(∠MFH)

Where students lose marks

Watch out: draw the altitude and write two Pythagoras equations, one per right triangle. Then combine them. Stopping at one equation loses the marks.

Part (b)(i)

9 more steps, each with the reason for it, in the full solution

Final answer

∠MFH = 52.08°

Where students lose marks

Watch out: find HF, MF and MH with Pythagoras first and keep them as exact surds. Rounding sides early gives a wrong angle.

Part (b)(ii)(a)

5 more steps, each with the reason for it, in the full solution

Final answer

Sketch (see video)

Where students lose marks

Watch out: use a ruler, join all three sides, and label H′, F′, M′ correctly. The obtuse angle belongs at F′.

Part (b)(ii)(b)

9 more steps, each with the reason for it, in the full solution

Final answer

∠F′M′H′ = 20.95°, area = 16.64 cm²

Where students lose marks

Watch out: don't round √61 and √142 early. It gives a wrong angle at H′, so the included angle and area go wrong too.

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