Properties of Triangles Questions with Solutions

15 questions3 easy12 medium

Properties of triangles connect the sides a, b, c and the opposite angles A, B, C of any triangle, not just right-angled ones. This topic covers the sine rule (a/sin A = b/sin B = c/sin C = 2R), the cosine rule (a^2 = b^2 + c^2 - 2bc cos A), the projection formulas, half-angle formulas, and formulas for the area of a triangle, including (1/2)ab sin C and Hero's formula. It also covers the circumradius R, the inradius r and their relationships with the area and semi-perimeter, such as r = area/s. Exam questions test applying these rules to find an unknown side, angle or area, and identifying the type of triangle from a given condition. Choosing between the sine rule and the cosine rule is usually the key decision. Practise the properties of triangles questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
Let $Δ A B C$ be a triangle whose area is $103$ units with side lengths $∣ A B ∣ = 8$ units and $∣ A C ∣ = 5$ units. Find possible values of the angle A.
A. $60^{\circ}$ or $120^{\circ}$ B. $45^{\circ}$ or $135^{\circ}$ C. $30^{\circ}$ onlyD. $90^{\circ}$ only
easyAsked in NIMCET 2013View solution ›
Q2.
In a $Δ A B C$ , $(c + a + b) (a + b - c) = ab$ . The measure of the angle C is
A. $\frac{π}{3}$ B. $\frac{π}{6}$ C. $\frac{2 π}{3}$ D. None of these
easyAsked in NIMCET 2013View solution ›
Q3.
If A, B and C are three angles of a $△ A B C$ , whose area is $Δ$ . Let a, b and c be the sides opposite to the angles A, B and C respectively. If $s = \frac{a + b + c}{2} = 6$ , then the product $\frac{1}{3} s^{2} (s - a) (s - b) (s - c)$ is equal to
A. $2Δ$ B. $2 Δ^{2}$ C. $2 Δ$ D. $Δ^{2}$
easyAsked in NIMCET 2014View solution ›