Sarvesh K Verma Solutions for Chapter: Geometry, Exercise 7: Test of Your Learning

In the following figure $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D}$ are the four vertices of the square and $\mathrm{E}, \mathrm{F}, \mathrm{G}$ and $\mathrm{H}$ are the four mid-points of the sides of the square $\mathrm{ABCD}.$ If each side of the square be $\mathrm{a},$ then how many of the following statements are correct about the octagon formed at the centre by joining the mid-points with the opposite vertices?

$\left(\mathrm{i}\right)$ It's an equiangular octagon
$\left(\mathrm{ii}\right)$ It's an equilateral octagon
$\left(\mathrm{iii}\right)$ It's a regular octagon

(IV) it's each side is $\frac{\sqrt{5a}}{12}$

(v)  it's each side is $\frac{a}{2\left(\sqrt{4 + 2\sqrt{2}}\right)}$

A straight line through the vertex of a triangle $\mathrm{PQR}$ intersects the side $\mathrm{QR}$ at the point $\mathrm{S}$ and the circumcircle of the triangle $\mathrm{PQR}$ at the point $\mathrm{T}.$ If $\mathrm{S}$ is not the centre of the circumcircle, then

$\left(\mathrm{i}\right) \frac{1}{\mathrm{PS}}+\frac{1}{\mathrm{ST}}<\frac{2}{\sqrt{\mathrm{QS}\times \mathrm{SR}}}$

$\left(\mathrm{ii}\right) \frac{1}{\mathrm{PS}}+\frac{1}{\mathrm{ST}}>\frac{2}{\sqrt{\mathrm{QS}\times \mathrm{SR}}}$

$\left(\mathrm{iii}\right) \frac{1}{\mathrm{PS}}+\frac{1}{\mathrm{ST}}<\frac{4}{\mathrm{QS}}$

$\left(\mathrm{iv}\right) \frac{1}{\mathrm{PS}}+\frac{1}{\mathrm{ST}}>\frac{4}{\mathrm{QR}}$

Let $\mathrm{ABCD}$ be a square of side length $2$ units. Let ${\mathrm{C}}_1$ be the incircle and ${\mathrm{C}}_2$ be the circumcircle of the square $\mathrm{ABCD}$. If $\mathrm{P}$ is a point on ${\mathrm{C}}_1$ and $\mathrm{Q}$ is a point on ${\mathrm{C}}_2$, the value of $\frac{{\mathrm{PA}}^2+{\mathrm{PB}}^2+{\mathrm{PC}}^2+{\mathrm{PD}}^2}{{\mathrm{QA}}^2+{\mathrm{QB}}^2+{\mathrm{QC}}^2+{\mathrm{QD}}^2}$ is

In a circle with centre $\mathrm{O},$ the diameter $\mathrm{PY}$ is perpendicular to chord $\mathrm{AB},$ as shown in the following figure. Find the correct relation between $\mathrm{DX}$ and $\mathrm{CY}.$

In the following diagram, a circle of radius $1/3$ unit is internally tangent to another circle of radius $1/4$ unit. The region between these two circles is shaded in black. Within the shaded region, there are numerous circles inscribed in such a way that each such triangle is tangent to at least $3$ other triangles. If there are total $25$ circles drawn in the shaded region, while maintaining the symmetricity, find the sum of curvatures of all the $25$ circles.

A circle ${\mathrm{C}}_1$ with diameter $1/25 \mathrm{cm}$ is tangent to $\mathrm{X}$-axis at $(2/5,0)$ and the other circle with diameter $1/49 \mathrm{cm}$ is tangent to $\mathrm{X}$-axis at $(3/7, 0).$ Moreover, these two circles are on the same side of $\mathrm{X}$-axis and tangent to each other. There is another circle ${\mathrm{C}}_3$ which is tangent to both of these circles and tangent to $\mathrm{X}$-axis. Find, the distance between the $\mathrm{Y}$-axis and the point of tangent of the circle ${\mathrm{C}}_3$ on $\mathrm{X}$-axis.

In the following diagram a semicircle with centre $\mathrm{O}$$O$ inscribes two semicircles with centers $\mathrm{P}$ and $\mathrm{Q},$ which are tangent to each other at a point $\mathrm{B}$ on the diameter $\mathrm{AC}$ of the larger semicircle. $\mathrm{AB}=\mathrm{p}$ and $\mathrm{BC}=\mathrm{q}$ are the diameters of the inscribed semicircles, respectively. A perpendicular $\mathrm{BD}$ is drawn from $\mathrm{D}$ to $\mathrm{B}. \mathrm{D}$ is a point on the largest semicircle. There are two circles with centre $\mathrm{M}$ and $\mathrm{N}$ inscribed on the different sides of $\mathrm{BD}$ such that these circles are tangent to $\mathrm{BD}$ as well as tangent to an inscribed semicircle and the largest semicircle. If the radii of these circles $\mathrm{M}$ and $\mathrm{N}$ be $\mathrm{m}$ and $\mathrm{n},$ respectively, which of the followings is/are correct?

$\left(\mathrm{i}\right) \mathrm{m}>\mathrm{n}$

$\left(\mathrm{ii}\right) \mathrm{m}=\mathrm{n}$

$\left(\mathrm{iii}\right) \mathrm{m}<\mathrm{n}$

$\left(iv\right) \mathrm{m}+\mathrm{n}=\frac{2\mathrm{pq}}{\mathrm{p}+\mathrm{q}}$

There are three circles inscribed in a triangle such that each circle is tangent to the other two circles and the two sides of the triangle. Let ${\mathrm{r}}_1,{\mathrm{r}}_2$ and ${\mathrm{r}}_3$ be the radii or circles ${\mathrm{C}}_1,{\mathrm{C}}_2$ and ${\mathrm{C}}_3,$ find the inradius of the triangle inscribing these three circles.