Properties of triangles practice · Medium

Option D (20 m) is correct because the minimum height is achieved when the angle is 45 degrees, resulting in a height of 20 m.

Option A (AP) is correct because if the cotangents of half the angles of a triangle are in AP, then the sides of the triangle are also in AP.

Option A ($\frac{\pi}{8}$) is correct because using trigonometric identities and the given ratio, we can derive that $\theta = \frac{\pi}{8}$.

Option C ($a=h(\sqrt{3}-1)$) is correct because it is derived by using trigonometric relationships and the sine rule in the given geometric setup.

Option D ($c\cot \frac{C}{2}$) is correct because after simplification using trigonometric identities, the expression reduces to this form.

Option A ($1 + \frac{r}{R}$) is correct because the expression $\cos A + \cos B + \cos C$ can be rewritten in terms of the inradius $r$ and circumradius $R$ as $1 + \frac{r}{R}$.

Option A ($1: 1: \sqrt{{2}}$) is correct because the given equation implies that the triangle is right-angled and isosceles, leading to the specified side ratio.

Option A ($4(3+\sqrt{3})$) is correct because the area of the triangle can be calculated using the formula $\frac{1}{2}ab\sin C$ after finding the angles and sides.

Option B ($\frac{1}{b}$) is correct because it simplifies to $\frac{1}{b}$ using projection formulas after substituting the cosine values.

Option A ($\frac{1}{2} b^2 \sin 2A$) is correct because the sum of the areas can be expressed in terms of $b$ and $A$ using trigonometric identities.

Option C ($c^2$) is correct because by expanding the given expression and using the cosine rule, it simplifies to $c^2$.

Option A (1) is correct because by substituting the angles and using trigonometric identities, the expression simplifies to 1.

Option C ($\frac{{{a^2} + {b^2} - 2}}{2}$) is correct because squaring and adding the given equations allows us to express $\sin(A+B)$ in terms of $a$ and $b$.

Option D ($\tan B\tan C = 1$) is correct because with A = 90°, B + C = 90°, making B and C complementary angles, and the product of their tangents is 1.

Option A ($\frac{b-c}{b+c}$) is correct because using trigonometric identities and the given right angle, the expression simplifies to this form.

Option A ($\frac{a}{2}$) is correct because using the sine rule and area of triangle formula, the altitude from A to BC is found to be $\frac{a}{2}$.

Option C ($\frac{\pi}{2}$) is correct because using the given condition $2r = a + c - b$ and relating it to the area of the triangle, we can deduce that angle B is 90 degrees.

Option C ($s-r$) is correct because in a right-angled triangle, the circumradius $R$ is related to the semi-perimeter $s$ and inradius $r$ by the formula $R = \frac{{s-r}}{{2}}$, which simplifies to $R = s - r$ when A=90.

Option D (2) is correct because the given cotangent relationships lead to the simplification of the expression to 2.

Option B ($4 \cos ^2 \frac{{A+C}}{{2}}$) is correct because by using the properties of arithmetic progressions and trigonometric identities, we can show that $4(1-\cos A)(1-\cos C)$ simplifies to $4 \cos ^2 \frac{{A+C}}{{2}}$.

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 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.

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.