Exercises from chapter 13 — Similarity of triangles and point powers

The triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$ in the figure are similar. If the similarity ratio of the 1st to the 2nd is $\dfrac{3}{2}$, determine: a) $a$, $b$ and $c$; b) the ratio between their perimeters.

Triangles $ABC$ and $PQR$ are similar. Determine $x$ and $y$.

If $\triangle KLM$ is similar to $\triangle FGH$, determine $x$.

The three sides of a triangle $ABC$ measure 8 cm, 18 cm and 16 cm. Determine the sides of a triangle $A^{\prime}B^{\prime}C^{\prime}$ similar to $ABC$, knowing that the similarity ratio of the first to the second is equal to 3.

If $\overline{DE}$ is parallel to $\overline{BC}$, determine $x$ in the cases (in b, $x = AD$):

From a $\triangle ABC$ we know that $AB = 20$ m, $BC = 30$ m and $AC = 25$ m. If $D$ is in $\overline{AB}$, $E$ in $\overline{AC}$, $\overline{DE}$ is parallel to $\overline{BC}$ and $DE = 18$ m, determine $x = DB$ and $y = EC$.

Show that, if the similarity ratio between two triangles is $k$, then the ratio between their perimeters is also $k$.

Two triangles $ABC$ and $A^{\prime}B^{\prime}C^{\prime}$ are similar. Knowing that the side $\overline{AB}$ measures 20 cm and that its counterpart $\overline{A^{\prime}B^{\prime}}$ measures 40 cm, determine the perimeter of the triangle $ABC$, knowing that the perimeter of the triangle $A^{\prime}B^{\prime}C^{\prime}$ is 200 cm.

The perimeter of a triangle is 60 m and one of the sides is 25 m. What is the perimeter of the similar triangle whose side opposite to the given side measures 15 m?

The sides of a triangle measure 8,4 cm, 15,6 cm and 18 cm. This triangle is similar to a triangle whose perimeter measures 35 cm. Calculate the longest side of the second triangle.

In a triangle $ABC$, the sides measure $AB = 4$ cm, $BC = 5$ cm and $AC = 6$ cm. Calculate the sides of a triangle similar to $ABC$ whose perimeter measures 20 cm.

A triangle whose sides measure 12 m, 18 m and 20 m is similar to one whose perimeter measures 30 m. Calculate the length of the smallest side of the smallest triangle.

In the figure, $AB = 2 \cdot BC$ and $BE = 14$. Calculate $CD$, knowing that $\overline{BE} \parallel \overline{CD}$.

The bases of a trapezoid measure 12 m and 18 m and the sides oblique to the bases measure 5 m and 7 m. Determine the sides of the smallest triangle that we obtain by extending the sides oblique to the bases.

If angles with “equal marks” are congruent, determine the unknowns in the cases:

If $\alpha = \beta$, determine $x$ and $y$ in the cases:

Determine $x$ and $y$ in the following cases:

Since $r$ and $s$ are parallel lines, determine $x$ in the cases:

If $\overline{AB} \parallel \overline{ED}$, $DE = 4$ cm, $CD = 2$ cm and $BC = 6$ cm, calculate the measurement of $\overline{AB}$.

In the figure, $\overline{AB}$ is parallel to $\overline{DE}$. a) Prove that the triangles $ABC$ and $EDC$ are similar. b) Being $AB = 5$, $AC = 6$, $BC = 7$ and $DE = 10$, calculate $CD$.

In the figure below, determine the value of $x$.

Calculate the value of $x$, knowing that the figure below is a parallelogram.

In the figure, $AB = 8$ cm, $BC = 3$ cm and $AE = 5$ cm. Calculate $x = DE$, knowing that $A\hat{C}E \equiv A\hat{D}B$.

In the figures, determine $x$.

Given the figure, determine the value of $x$.

In the figure, let us consider the squares with sides $a$ and $b$ ($a > b$). Calculate the value of $x$.

In the figure, let us consider the squares with sides $x$, 6 and 9. Determine the perimeter of the square with side $x$.

Determine the side length of the square in the figure below.

Extending the oblique sides to the bases of a trapezoid, we obtain a point $E$ and the triangles $ECD$ and $EAB$. Determine the relationship between the heights of the two triangles, relative to the sides that are the bases of the trapezoid, with 12 cm and 4 cm being the measurements of the bases of the trapezoid.

The bases of a trapezoid $ABCD$ measure 50 cm and 30 cm, and the height, 10 cm. Extending the non-parallel sides, they intersect at a point $E$. Determine the height $\overline{EF}$ of the triangle $ABE$ and the height $\overline{EG}$ of the triangle $CDE$.

In a isosceles triangle of 20 cm in height and $\dfrac{50}{3}$ cm in base, a rectangle of 8 cm in height is inscribed at the base of the triangle. Calculate the measurement of the base of the rectangle.

Determine $x$ and $y$.

In the figure, we have $AB = 8$, $BC = 15$, $AC = 17$ and $EC = 4$. Determine $x = DE$ and $y = CD$.

In the figure, the square $DEFG$ is inscribed in the triangle $ABC$. With $BD = 8$ cm and $CE = 2$ cm, calculate the perimeter of the square.

In a rectangle $ABCD$, the sides $\overline{AB}$ and $\overline{BC}$ measure 20 cm and 12 cm. With $M$ being the midpoint of the side $\overline{AB}$, calculate $EF$, the distance from the point $E$ to the side $\overline{AB}$, with $E$ being the intersection of the diagonal $\overline{BD}$ with the segment $\overline{CM}$.

In the figure, determine $x$.

Let us consider a triangle $ABC$ with side $BC = 10$ cm. Let $\overline{CD}$ be an internal segment of the triangle, with $D$ on the $\overline{AB}$ side. Knowing that $BD = 4$ cm and that the angles $B\hat{A}C$ and $B\hat{C}D$ are congruent, determine the measure of $\overline{AD}$.

Through the points $A$ and $B$ of a straight line, perpendiculars are drawn to the straight line. On them we take the segments $AC = 13$ cm and $BD = 7$ cm. In the segment $AB = 25$ cm, a point $P$ is taken such that the angles $A\hat{P}C$ and $B\hat{P}D$ are congruent. Calculate the measurement of $\overline{AP}$.

Consider the circle circumscribed by a triangle $ABC$. Let $\overline{AE}$ be the diameter of this circle and $\overline{AD}$ be the height of the triangle. Being $AB = 6$ cm, $AC = 10$ cm and $AE = 30$ cm, calculate the height $\overline{AD}$.

Calculate $R$, the radius of the circle circumscribed by the triangle $ABC$ in the figure, being $AB = 4$, $AC = 6$ and $AH = 3$.

Two circles of radii $R$ and $r$ are tangent externally at the point $A$. Since $C$ and $D$ are the points of tangency of a straight line $t$ external to the two circles, determine the height of the triangle $ACD$ relative to the side $\overline{CD}$.

The point $O$ is the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$ of a rhombus $ABCD$. The side $\overline{AD}$ is extended to a point $F$ so that $DF = 4$ m. If $\overline{OF}$ meets $\overline{CD}$ in $E$ and $ED = 2$ m, determine the side of the diamond.

From a triangle $ABC$ we know that the angle $\hat{A}$ is twice the angle $\hat{C}$, $AB = 6$ m and $AC = 10$ m. Determine $BC$.

In the figure, the rays $\overrightarrow{PA}$ and $\overrightarrow{PB}$ are tangents to the circle. If the distances from $Q$ to the tangents are 4 and 9, find the distance from $Q$ to the chord $\overline{AB}$.

The lines $t$ and $f$ are tangents to the circles in $A$. Determine $AB$ in terms of $a = BC$ and $b = BD$.

In the figure, $\overline{RQ}$ is perpendicular to $\overline{PQ}$, $\overline{PQ}$ is perpendicular to $\overline{PT}$, and $\overline{TS}$ is perpendicular to $\overline{PR}$. Prove that $TS \cdot RQ = PS \cdot PQ$.

In each case, determine the unknown.

Determine the value of $x$ in the figures below.

Determine $x$ in cases:

In the figure, calculate the measurements of the strings $\overline{BD}$ and $\overline{CE}$, being $AB = 3x$, $AC = 4x - 1$, $AD = x + 1$ and $AE = x$.

Determine the value of $x$ in the figures below.

Determine the radius of the circle in the following cases:

In the figure, being $ED : EC = 2 : 3$, $AE = 6$ and $EB = 16$, calculate the length of $\overline{CD}$.

Determine the measurement of the segment $\overline{DE}$, knowing that $\overline{AB}$ is the diameter of the circle, $B$ is the point of tangency of the segment $\overline{BC}$ to the circle and $\overline{DE}$ is parallel to $\overline{BC}$.

Calculate the power of a point $P$ in relation to a circle with center $O$ and radius $r$, as a function of the distance $d$ between $O$ and $P$ and the radius $r$.

In the figure in the book, calculate $\text{pot}\,A + \text{pot}\,B + \text{pot}\,C$ (powers in relation to the circumference $\lambda$).

Through a point $P$ distant 18 cm from a circle, a secant is drawn that determines a chord $\overline{AB}$ measuring 10 cm on the circle. Calculate the length of the tangent to this circle drawn from the point $P$, knowing that $\overline{AB}$ passes through the center of the circle.

Determine the radius of the smaller circle inscribed in a quadrant of the larger circle in the figure, with $2R$ being the diameter of the larger circle.

Two chords $\overline{AB}$ and $\overline{CD}$ intersect at a point $P$ inside a circle. Determine the measurement of the segment $\overline{BP}$, knowing that the segments $\overline{CP}$, $\overline{DP}$ and the chord $\overline{AB}$ measure, respectively, 1 cm, 6 cm and 5 cm.

In a circle, two ropes cut each other. The product of the two segments of the first string is $25\ \text{cm}^2$. It is known that, in the second string, the smallest segment is worth $\dfrac{1}{4}$ of the largest. Determine the length of the longest segment of this second string.

$\overline{AB}$ and $\overline{AC}$ are two strings of equal measurements, belonging to a circle. A chord $\overline{AD}$ intersects the chord $\overline{BC}$ at a point $P$. Prove that the triangles $ABD$ and $APB$ are similar.