Mathematics

Mathematics

Problem

A student tries to construct a triangle with PQ=5 cmPQ = 5\,\text{cm}, QR=4 cmQR = 4\,\text{cm} and ∠P=30∘\angle P = 30^{\circ}. He finds that two different positions for point RR are possible. Which of the following best explains this outcome? a) The data satisfy the SSS condition, allowing two triangles. b) The data satisfy the SAS condition, so two different triangles can be drawn. c) The data give an SSA set, which does not determine a unique triangle. d) The data give an ASA set, which always produces two triangles.
Intuitive explanation

This is about the ambiguous case of triangle construction (SSA), where two sides and a non-included angle don't uniquely determine a triangle. When you fix side PQPQ and angle PP, point RR must lie on a ray from PP at 30°30°; simultaneously, RR must be exactly 4 cm4\,\text{cm} from QQ. A circle of radius 4 cm4\,\text{cm} centered at QQ can intersect this ray at two different points, creating two valid triangles—this happens when the given measurements fall in the "ambiguous zone" where the circle cuts the ray twice.

Intuitive explanation

This is about the ambiguous case of triangle construction (SSA), where two sides and a non-included angle don't uniquely determine a triangle. When you fix side PQPQ and angle PP, point RR must lie on a ray from PP at 30°30°; simultaneously, RR must be exactly 4 cm4\,\text{cm} from QQ. A circle of radius 4 cm4\,\text{cm} centered at QQ can intersect this ray at two different points, creating two valid triangles—this happens when the given measurements fall in the "ambiguous zone" where the circle cuts the ray twice.

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