Chapter exercises 10 — Circumference and circle

Determine the radius of the center circle $O$, given $AB = 3x - 3$ and $OA = x + 3$.

The circle has a radius of 16 cm and the point $P$ is 7 cm from the center. Determine the distance between $P$ and the circumference.

Determine the value of $x$ in the cases: a) $s$ is perpendicular to $\overline{AB}$; b) $\overline{PA}$ and $\overline{PB}$ are tangents to the circle.

Determine the value of $x$, with $O$ being the center of the circle, in the following cases:

The circumferences of the figure are externally tangent. If the distance between the centers is 28 cm and the difference between the radii is 8 cm, determine the radii.

Two circles are internally tangent and the sum of their radii is 30 cm. If the distance between the centers is 6 cm, determine the radii.

In the figure, the circles are tangents two by two and the centers are the vertices of the triangle $ABC$. Being $AB = 7$ cm, $AC = 5$ cm and $BC = 6$ cm, determine the radii of the circles.

The circles are externally tangent to $Q$, and $\overrightarrow{PA}$ and $\overrightarrow{PB}$ are tangents to the circles. Determine the measure of the angle $A\hat{Q}B$ in the cases: a) where $t$ is a common tangent and $A\hat{P}B = 80^{\circ}$; b) with $A\hat{P}B = 100^{\circ}$.

State the number of straight lines that pass through the point $P$ and touch the circle $\lambda$ in the following cases: a) $P$ belongs to $\lambda$; b) $P$ is inside $\lambda$; c) $P$ is external to $\lambda$.

Determine the number of common tangent lines that we can draw to two circles in the cases: a) different concentric; b) exterior; c) drying agents; d) tangents externally; e) tangents internally.

Can a circular sector coincide with a circular segment? Name the case.

In which case is a circular sector a semicircle?

What can we say about the line that passes through the point of tangency of two circles tangent to each other, knowing that this line is perpendicular to the line that passes through the centers?

Is it possible to obtain a chord that passes through the midpoint of the diameter of a circle?

Give the position of two circles with radii $r$ and $R$, with $d$ being the distance between their centers, in the cases: a) $r = 2$ cm, $R = 5$ cm, $d = 10$ cm; b) $r = 5$ cm, $R = 10$ cm, $d = 15$ cm; c) $r = 3$ cm, $R = 7$ cm, $d = 4$ cm; d) $r = 6$ cm, $R = 10$ cm, $d = 0$ cm; e) $r = 6$ cm, $R = 8$ cm, $d = 10$ cm.

The distance between the centers of two outwardly tangent circles is 33 cm. Determine their diameters, knowing that the ratio between their radii is $\dfrac{4}{7}$.

The distance between the centers of two internally tangent circles is 5 cm. If the sum of the radii is 11 cm, determine the radii.

Two circles are secants, with 20 cm being the distance between their centers. Knowing that the radius of the smallest circle measures 11 cm, determine the radius of the largest, which is a multiple of 6.

Two circles with centers $A$ and $B$ are externally tangent and internally tangent to a circle with center $C$. With $AB = 12$ m, $AC = 17$ m and $BC = 13$ m, determine the radii of these circles.

Let $P$ be the point of tangency of the circle inscribed in the triangle $ABC$ with the side $\overline{AB}$. If $AB = 7$, $BC = 6$ and $AC = 8$, how much is $AP$ worth?

Consider a triangle $ABC$ with sides $AB = c$, $AC = b$ and $BC = a$, and let $P$, $Q$ and $R$ be the points at which the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ touch the inscribed circle. Calculate the segments $AR = x$, $BP = y$ and $CQ = z$.

In the figure, determine the measurement of the segment $\overline{BD}$, knowing that the circle with center $O$ is inscribed in the triangle $ABC$ and that the sides $\overline{AB}$, $\overline{BC}$ and $\overline{AC}$ measure 6 cm, 8 cm and 10 cm, respectively.

In the figure, the center circle $O$ is inscribed in the triangle $ABC$, with $BD = 4$, $AF = 3$ and $EC = 5$. What is the perimeter of triangle $ABC$?

In the figure, knowing that $AB = c$, $BC = a$, $AC = b$ and $p$ is the semiperimeter of the triangle, prove that $AP$ is equal to $p - a$.

A circle is inscribed in a triangle $ABC$ and touches the sides $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$, respectively, in $P$, $Q$ and $R$. If $AB = c$, $AC = b$, $BC = a$ and the semiperimeter is $p$, calculate $AR$, $BP$ and $CQ$.

In the figure, $PA = 10$ cm. Calculate the perimeter of the triangle $PRS$.

In the figure, $\overline{PA}$ is equal to three times the diameter of the circle. Determine the measurement of the perimeter of the triangle $PDE$ as a function of the radius $r$ of that circle.

The hypotenuse of a right-angled triangle measures 10 cm and the radius of the inscribed circle measures 1 cm. Calculate the perimeter of the triangle.

See the resolution of the book

The quadrilateral $SATO$ is a square with side 1 cm. Indicating $BP = BT = a$ and $CP = CS = b$:

$$2p = (a + 1) + (b + 1) + (a + b) = 2(a + b) + 2 = 2 \cdot 10 + 2 = 22 \text{ cm}.$$

In the figure, calculate the measurement of the radius $r$ of the circle inscribed in the right-angled triangle $ABC$, being $AB = 10$ cm, $AC = 24$ cm and $BC = 26$ cm.

Determine the diameter of a circle inscribed in a right-angled triangle whose sides measure 9 cm, 12 cm and 15 cm.

Determine the radius of a circle inscribed in a right-angled triangle with legs $b$ and $c$ and hypotenuse $a$.

In the figure, with $2p = a + b + c$ and $r$ being the radius of the inscribed circle, calculate the measure of the hypotenuse $a$ as a function of $p$ and $r$.

Determine the perimeter of the circumscribed quadrilateral $ABCD$ of the figure, with $AB = 3x + 1$, $BC = 2x$, $CD = x + 1$ and $DA = 3x$.

$ABCD$ is a circumscriptible quadrilateral whose sides measure $AD = 12$ cm, $DC = 9$ cm, $BC = x + 7$ and $AB = 2x + 1$. Determine the perimeter of this quadrilateral.

Calculate the value of the radius $r$ of the circle inscribed in the rectangular trapezoid in the figure.

The difference of two opposite sides of a circumscribable quadrilateral is equal to 8 cm, and the difference of the other two sides is 4 cm. Determine the sides of the quadrilateral, with 56 cm being their sum.

In the figure, determine the perimeter of the triangle $ADE$, knowing that the perimeter of the triangle $ABC$ is worth 10 cm, the base $\overline{BC}$ measures 4 cm and the circle is inscribed in the quadrilateral $BCDE$.

Determine the measurement of one of the non-parallel sides of an isosceles trapezoid circumscribed by a circle, knowing that its bases measure 30 cm and 10 cm.

Prove that any parallelogram circumscribed by a circle is a rhombus.

Prove that the diameter is the largest chord of a circle.

Prove that if two chords on a circle are the same distance from the center, then they are congruent.