Properties of triangles practice · Mixed difficulty

Option B ($\frac{{a^4-b^4}}{{a^2 b^2}}$) is correct because it results from substituting the sine and cosine ratios in terms of sides a and b and simplifying the expression.

Option A is correct because using the sine rule and the formula for the area of a triangle, we can calculate the area to be $\frac{1}{{\sqrt 3 - 1}}$.

Option C (c²) is correct because the expression simplifies to c² using trigonometric identities and the cosine rule.

Option B (2) is correct because the trigonometric expression simplifies to 2 using trigonometric identities and the fact that A, B, and C are angles of a triangle.

Option C ($\frac{{\sqrt 3 }}{2}\,\;\mathrm{sq}.\mathrm{\;unit}$) is correct because the area of the triangle $ABC$ with $a = 1$, $b = 2$, and $\angle C = 60^\circ$ is $\frac{{\sqrt 3 }}{2}\,\;\mathrm{sq}.\mathrm{\;unit}$.

Option C ($3{c^2}a$) is correct because using the sine rule and trigonometric identities, it can be shown that ${a^3} + {b^3} = 3a{c^2}$ when AB=AC and angle A is 20 degrees.

Option A ($\frac{\sqrt{3}-1}{2}$) is correct because it is the correct inradius calculated using the given side lengths and angle.

Option C ($120^{\circ}$) is correct because the largest angle can be found using the cosine rule, which results in an obtuse angle of $120^{\circ}$.

Option A (either isosceles or right angled) is correct because the given equation simplifies to either A = B (isosceles) or A² + B² = C² (right-angled).

Option A (An equilateral triangle) is correct because an equilateral triangle with integer coordinates cannot exist due to area considerations.

Option C ($1/r$) is correct because the sum of the reciprocals of the exradii is equal to the reciprocal of the inradius, which is a standard result in triangle geometry.

Option C (Both (1) and (2)) is correct because both statements regarding the relationships between angles/sides and exradii/sides in a triangle are true.

Option D ($\frac{1}{r} - \frac{1}{{2R}}$) is correct because it is the simplified form of the given expression using trigonometric identities and the properties of triangles.

Option C (AP) is correct because if the altitudes are in HP, then the sides are in AP, and therefore the sines of the angles are also in AP.

Option C ($\theta \in(\frac{\pi}{12}, \frac{\pi}{6})$) is correct because the angle $\theta$ opposite to the base is acute and falls within the specified interval after solving the trigonometric equations.

Option C ($9\frac{1}{2}$) is correct because by pairing terms using the identity sin²θ + cos²θ = 1 and accounting for the remaining terms, the sum equals 9.5.

Option D ($4\;\Delta$) is correct because it is the simplified form of the expression after applying the sine rule and the area formula of a triangle.

Option B (I is true, II is false) is correct because the first statement is a known trigonometric identity, while the second statement's implication is not universally true.

The correct answer is A (AP). Using trigonometric identities and the given equation, we can show that $a+c=2b$, which implies that the sides are in Arithmetic Progression.

Option B ($3:5:7$) is correct because the ratio of the sides can be found by setting up equations based on the given conditions that the sides are in A.P. and the area is 3/5 times the area of an equilateral triangle of the same perimeter.

Option A $\sin A + \sin B + \sin C$ is correct because $\frac{s}{R}$ can be expressed as $\frac{{a + b + c}}{{2R}}$, which is equal to $\sin A + \sin B + \sin C$ using the sine rule.

Option C ($4, 3$) is correct because by using the given equations and the Pythagorean theorem, we can solve for the values of $a$ and $c$.

Option A $\sqrt a + \sqrt b > \sqrt c$ is correct because it follows from the triangle inequality theorem.

Option D is correct because by applying the Law of Cosines and simplifying the given expression, the result equals 1.

Option A (1) is correct because simplifying the expression using trigonometric identities and the sine rule results in 2r/R, which has a maximum value of 1.

Option B (the altitudes are in HP) is correct because if $\sin A, \sin B, \sin C$ are in AP, then $a, b, c$ are in AP, and the altitudes, being inversely proportional to the sides, are in HP.

Option A ($120^{\circ}$) is correct because the greatest angle is found using the cosine rule with the longest side of the triangle.

Option A (right angled) is correct because the given condition leads to the Pythagorean theorem, indicating a right-angled triangle.

Option C ($\frac{{\sqrt 3 }}{2}\,\mathrm{sq }\,\mathrm{unit}$) is correct because the area of a triangle can be found using the formula $\frac{1}{2}ab\sin C$ when two sides and the included angle are known.

Option B ($\sin ^2 A$) is correct because the given expression simplifies to $\sin^2 A$ using Heron's formula and the area of the triangle.