Geometry

If $\triangle ABC \sim \triangle PQR$, $area(\triangle ABC) = 9$ cm$^2$, $area(\triangle PQR) = 16$ cm$^2$ and $BC = 2.1$ cm, find $QR$.

Two isosceles triangles have equal vertical angles and their areas are in the ratio $16:25$. Find the ratio of their corresponding heights.

In the figure, $DE \parallel BC$. If $AD = x$, $DB = x-2$, $AE = x+2$ and $EC = x-1$, find the value of $x$.

The areas of two similar triangles are $81$ cm$^2$ and $49$ cm$^2$. If the altitude of the larger triangle is $4.5$ cm, find the corresponding altitude of the smaller triangle.

In the diagram, $\triangle ABC \sim \triangle DEF$. If the ratio of their corresponding medians is $4:7$, what is the ratio of their areas?

In the figure, $AB \parallel CD \parallel EF$. If $AB = 6$ cm, $EF = 10$ cm, find the length of $CD$.

In $\triangle ABC$, $D$ and $E$ are points on $AB$ and $AC$ such that $DE \parallel BC$. If $Area(\triangle ADE) = 16$ cm$^2$ and $Area(\triangle ABC) = 25$ cm$^2$, and $DE = 4$ cm, find $BC$.

In the given figure, $ABCD$ is a trapezium with $AB \parallel DC$. The diagonals $AC$ and $BD$ intersect at point $O$. If $AO = (3x - 1)$ cm, $OC = (5x - 3)$ cm, $BO = (2x + 1)$ cm, and $OD = (6x - 5)$ cm, calculate the value of $x$.

A civil engineer is designing a ramp alongside a staircase. The staircase follows a straight line from point $A$ to point $B$. A vertical support beam $PQ$ is placed such that $P$ lies on the base $AC$ and $Q$ lies on the hypotenuse $AB$. If $AC = 12$ m, $BC = 5$ m, and the distance $AP = 4$ m, find the length of the support beam $PQ$ given that $PQ$ is perpendicular to $AC$.

In the given figure, $DE \parallel BC$ and $AD:DB = 5:4$. Find the ratio of $Area(\triangle ADE) : Area(\triangle ABC)$.

In the given figure, $O$ is the center of the circle. $\angle OAB = 35^{\circ}$. Find $\angle ACB$.

A point $P$ is $25$ cm from the center $O$ of a circle of radius $7$ cm. Find the length of the tangent $PT$ to the circle.

In the figure, $\angle DAC = 30^{\circ}$ and $\angle BAC = 45^{\circ}$. Find $\angle DBC$.

If the length of a chord of a circle is $16$ cm and it is at a distance of $6$ cm from the center, find the radius of the circle.

In a circle with center $O$, $\angle AOB = 100^{\circ}$. Find the angle between the tangents at $A$ and $B$.

In the figure, $O$ is the center of the circle. $AB$ is a chord and $AT$ is the tangent at $A$. If $\angle BAT = 50^{\circ}$, find $\angle ACB$ where $C$ is a point on the major arc.

Two circles touch each other externally at $C$. $AB$ is a common tangent. Find the measure of $\angle ACB$.

A circle is inscribed in a $\triangle ABC$ touching $AB, BC$ and $CA$ at $P, Q$ and $R$ respectively. If $AB = 10$ cm, $BC = 12$ cm and $CA = 8$ cm, find the length of $AR$.

Two circles of radii $5$ cm and $3$ cm intersect at two points. If the distance between their centers is $4$ cm, find the length of the common chord.

In the figure, $AB$ is a diameter of the circle with center $O$. $CD$ is a chord equal to the radius of the circle. If $AC$ and $BD$ are extended to intersect at $P$, calculate $\angle APB$.

The number of tangents that can be drawn from a point inside a circle is:

If $P$ is an external point and $PA$ and $PB$ are tangents to a circle with center $O$, then triangle $PAB$ is always:

When constructing an inscribed circle of a regular hexagon, the center of the circle is the intersection of its:

To draw a tangent at a point $A$ on a circle using its center $O$, the tangent line must be:

If two tangents $PA$ and $PB$ are drawn to a circle with center $O$ such that $\angle AOB = 110^\circ$, then $\angle APB$ is equal to:

A circle is inscribed in a triangle with sides $8, 15,$ and $17$. What is the radius of the incircle?

In triangle $ABC$, the incircle touches $AB, BC,$ and $CA$ at $D, E,$ and $F$ respectively. If $AD = 4$ cm, $BE = 5$ cm, and $CF = 6$ cm, find the perimeter of $\triangle ABC$.

A circular ring of radius $R$ is placed such that it touches both the x-axis and the y-axis in the first quadrant. A smaller ring of radius $r$ is placed in the corner such that it touches both axes and the larger ring externally. What is the ratio $R/r$?

A student constructs an isosceles triangle $ABC$ with $AB = AC = 10$ cm and $BC = 12$ cm. He then constructs the incircle. What is the distance from the vertex $A$ to the point of contact of the incircle with side $AB$?

Consider a circle with center $O$ and radius $5$ cm. From an external point $P$, two tangents $PA$ and $PB$ are drawn. If the angle between the tangents $\angle APB = 60^{\circ}$, what is the length of the chord $AB$?