Chapter exercises 14 — Right triangles

Determine the value of $x$ in the following cases:

Determine $x$ as a function of $a$ in the following cases:

Determine $x$ in cases:

Determine $x$ in cases:

Write 10 metric relations with the elements indicated in the figure.

Determine $x$ in cases:

In the figure, determine the elements $x$, $y$, $z$ and $t$.

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

Determine the value of $x$.

Calculate the elements $y$, $z$, $t$ and $x$ in the figure.

Determine the radius of the circle in the following cases:

Determine $x$ in the cases: a) isosceles triangle; b) equilateral triangle.

Determine the value of $x$ in the cases: a) rectangle; b) square.

Determine the value of $x$ in isosceles trapezoids.

Determine the value of $x$ in the rectangular trapezoids.

Determine the value of $x$ in the diamonds.

Determine the value of $x$ in parallelograms.

Determine the height of the trapezoid in the figure.

Determine the value of $x$ in the cases.

Determine the value of $x$ in the following cases:

See the resolution of the book (item d)

Tracing the rays that go to the contact points, we obtain a rectangular trapezoid whose height is $x$. Applying Pythagoras’ theorem to the highlighted triangle:

$$x^2 + 1^2 = 8^2 \;\Rightarrow\; x^2 = 63 \;\Rightarrow\; x = 3\sqrt{7}.$$

Determine the radius of the circle in the following cases:

Determine the value of $x$ in the cases (in b, $AB = 15$):

Determine the radius of the circle in the figures: a) rectangular trapezoid with bases 10 m and 15 m; b) $AH = 25$ m, $BC = 30$ m and $AB = AC$.

Determine the value of $x$ in the following cases:

Determine the radius of the circle, in the cases, if the right-angled triangle has: a) sides of 6 m and 8 m; b) a side of 8 m and hypotenuse of $4\sqrt{13}$ m.

Determine the value of $x$ in the figures:

Determine the diagonal of a square with perimeter 20 m.

Determine the diagonal of a rectangle with perimeter 20 m and base 6 m.

The perimeter of a rhombus is 52 m and a diagonal measures 10 m. Calculate the other diagonal.

Determine the height of a perimeter equilateral triangle 24 m.

Determine the perimeter of a height equilateral triangle 6 m.

The perimeter of a isosceles triangle is 18 m and the height relative to the base measures 3 m. Determine the base.

Determine the smallest height of a triangle whose sides measure 4 m, 5 m and 6 m.

Determine the height not relative to the base of a isosceles triangle with sides 10 m, 10 m and 12 m.

The height of a rectangle measures 8 m and the diagonal exceeds the base by 2 m. Calculate the diagonal.

The perimeter of a rectangle is 30 m and the diagonal measures $5\sqrt{5}$ m. Determine the sides of this rectangle.

The height relative to the base by isosceles triangle exceeds the base by 2 m. Determine the base if the perimeter is 36 m.

Each of the congruent sides of a isosceles triangle exceeds the base by 3 m. Determine the base if the height relative to it is 12 m.

The difference between the measurements of the diagonals of a rhombus with 68 m perimeter is 14 m. Determine the diagonals of this rhombus.

The bases of a rectangular trapezoid measure 3 m and 9 m and its perimeter is 30 m. Calculate the height.

Calculate the height and projections of the legs on the hypotenuse, in the right-angled triangle of legs 12 cm and 16 cm.

Calculate the hypotenuse, the height relative to the hypotenuse and the projections of the legs on the hypotenuse of a right-angled triangle of legs 3 and 4.

Given a equilateral triangle side $a$, calculate its height.

A ladder 2,5 m high rests against a wall and its foot is 1,5 m away from the wall. Determine how high the ladder reaches the wall under these conditions.

The height relative to the hypotenuse of a right-angled triangle measures 12 m. If the hypotenuse measures 25 m, calculate the legs.

In a triangle $ABC$, a rectangle in $A$, the height relative to the hypotenuse measures 1,2 cm and the hypotenuse measures 2,5 cm. Being $m$ and $n$, respectively, the projections of the largest and smallest sides on the hypotenuse, calculate $\dfrac{m}{n}$.

Two cyclists leave the same city in a straight direction; one towards the east and the other towards the north. Determine the distance that separates them after two hours, knowing that the cyclists’ speed is 30 km/h and 45 km/h, respectively.

The bases of an isosceles trapezoid measure 12 m and 20 m, respectively. The sum of the non-parallel sides is equal to 10 m. How tall is it?

The bases of an isosceles trapezoid measure 7 and 19 and the non-parallel sides, 10. Calculate the height of this trapezoid.

In a rectangular trapezoid, the sum of the bases is 16 cm, one of which is the $\dfrac{3}{5}$ of the other. Determine the height, knowing that the oblique side measures 5 cm.

In the figure, calculate the height of the rectangular trapezoid $ABCD$.

Knowing that the sum of the squares of the legs and the square of the hypotenuse of a right-angled triangle is equal to 200, determine the measure of the hypotenuse of that triangle.

Calculate the perimeter of isosceles triangle of 16 cm in base and 6 cm in height.

Determine the height of a trapezoid with bases 24 cm and 10 cm, knowing that the non-parallel sides measure 15 cm and 13 cm respectively.

The largest base and one of the sides oblique to the bases of an isosceles trapezoid circumscribing a circle are respectively equal to 18 cm and 13 cm. Determine the height of the trapezoid.

A chord common to two secant circles measures 16 cm. With 10 cm and 17 cm measuring the radii of the circles, determine the distance between their centers.

Let it be a point $P$, external to a circle. The shortest distance from this point to the circumference is 6 cm and the greatest distance from this point to the circumference is 24 cm. Determine the length of the tangent segment to the circle through this point.

Two circles of radii 12 cm and 20 cm are externally tangent. Determine the length of the segment $\overline{PQ}$, a tangent common to the two circles, with $P$ and $Q$ points of tangency.

A circumscriptible isosceles trapezoid has bases measuring 8 cm and 16 cm. Calculate the height of the trapezoid.

Prove that the diameter of a circle inscribed in an isosceles trapezoid is the geometric mean between the bases of the trapezoid.

Calculate the measurement of the radius of the circle, in the figure, knowing that $AD = 12$ cm, $AE = 15$ cm and $AB = 8$ cm.

In a isosceles triangle of height 8, a circle of radius 3 is inscribed. Calculate the measurement of the base of the triangle.

On the hypotenuse $\overline{AB}$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed, with hypotenuse $\overline{AB}$. If $BC = 1$, $AC = b$ and $AD = 2$, calculate $BD$.

In the trapezoid $ABCD$ in the figure, the diagonal $\overline{AC}$ is perpendicular to the oblique side $\overline{AD}$. With $CD = 25$ cm and $AD = 15$ cm, determine the height of the trapezoid.

Determine the diagonal measurement $\overline{AC}$ of the rectangular trapezoid in the figure, knowing that the bases measure respectively 4 cm and 9 cm and that the side $\overline{BC}$ measures $\sqrt{34}$ cm.

The segment $\overline{AB}$ has its ends $A$ and $B$ as points of tangency to the circumferences of centers $O_1$ and $O_2$. With 15 cm and 3 cm being the radii of these circles, respectively, and 24 cm being the distance between their centers, determine the segment $\overline{AB}$.

Determine the measure of the hypotenuse of a right-angled triangle, with 24 m being its perimeter and $\dfrac{24}{5}$ m being the measure of the height relative to the hypotenuse.

Consider a semicircle of diameter $AOB = 2r$. We internally construct two new semicircles with diameters $\overline{OA}$ and $\overline{OB}$ and a circle tangent to these three semicircles. Calculate the radius of this circle.

On the same side of a straight line, three circles are drawn tangent to the straight line and tangent to each other two by two. Knowing that two of them have a radius equal to 16, calculate the radius of the third.

A regular octagon is formed by cutting isosceles right triangles at the vertices of a square. If the side of the square measures 1, how long are the sides of the removed triangles?

Let us consider two tangent circles as in the figure. With $E$ being the center of the smaller circle, $F$ the point of tangency between the two circles and $a$ the side of the square, determine the radius of the smaller circle as a function of $a$.

Consider a square $Q$ with side $a$ and five circles of the same radius $r$ interior to $Q$, of which one is concentric with $Q$ and tangent externally to the four others, and each of these is tangent to two consecutive sides of $Q$. Determine the measurement of $r$ as a function of the measurement $a$ of the side of the square.

In the figure, determine the radius of the circle, knowing that $\overline{AC}$ and $\overline{AD}$ tangent to the circle at points $C$ and $D$, respectively, and that $BE = 12$ cm and $AE = 54$ cm.

Two cable cars $T_1$ and $T_2$ leave from a station $E$ located on a horizontal plane, towards the peaks $P_1$ and $P_2$ of two mountains. Determine the distance between $P_1$ and $P_2$, knowing that the cable cars covered 1 500 m and 2 900 m, respectively, that the first mountain is 900 m high and the second 2 000 m and that the feet of the mountains and $E$ are in a straight line.

Let there be two circles tangent to each other, internally, as in the figure. Being $PQ = 8$ cm and $ST = 3$ cm, calculate the measurement of $\overline{RQ}$.

In a circle with center $O$ and radius $R$, a chord $AB = \dfrac{R}{2}$ is considered. Calculate the radius of the circle inscribed in the circular sector $OAB$.

On the sides of a square, we draw externally four isosceles triangles with heights relative to the bases equal to 3 cm. Determine the perimeter of the square, knowing that the vertices of the four triangles belong to the same circle, with a radius equal to $3(\sqrt{2} + 2)$ cm.

Two squares $ABCD$ and $CDEF$ have the side $\overline{CD}$ in common. We draw the diagonals $\overline{AC}$ and $\overline{EC}$. Being $AM = \dfrac{1}{3}AC$ and $EP = \dfrac{1}{2}CE$, with $M$ in $\overline{AC}$ and $P$ in $\overline{CE}$, determine the segment $\overline{PM}$, depending on the side $a$ of the squares.

Determine the distance between the feet of the height and median relative to the hypotenuse of a right-angled triangle with $(18 + 6\sqrt{3})$ m in perimeter, knowing that the projections of the legs on the hypotenuse are directly proportional to the numbers 1 and 3.

Determine the perimeter of a right-angled triangle, knowing that the median and height relative to the hypotenuse measure, respectively, 4 cm and $2\sqrt{3}$ cm.

Given the right-angled triangle $ABC$ with legs $\overline{AB}$ and $\overline{AC}$ respectively equal to 80 cm and 60 cm, consider the height $\overline{AH}$ and the median $\overline{AM}$ relative to the hypotenuse of the triangle. Calculate the measurements of the segments $\overline{AH}$, $\overline{AM}$, $\overline{HB}$, $\overline{HC}$, $\overline{MH}$, as well as the hypotenuse of the triangle.

Determine the height relative to the base of a isosceles triangle as a function of the base $a$ and the radius $r$ of the inscribed circle.

Determine the internal angle bisector relative to the hypotenuse of a right-angled triangle with legs $b$ and $c$.

See the resolution of the book

Let $x$ be the measurement of angle bisector $\overline{AS}$ relative to the hypotenuse. Through $S$ we draw a segment $\overline{SP}$ parallel to one of the legs — parallel to $b$, for example. The triangle $APS$ is rectangular and isosceles ($A\hat SP = 45^{\circ}$), with legs $y$:

Triangles $BAC$ and $BPS$ are similar. Then:

$$\begin{cases} y^2 + y^2 = x^2 \\ \dfrac{b}{y} = \dfrac{c}{c - y} \end{cases} \Rightarrow \begin{cases} y = \dfrac{x}{\sqrt{2}} \\ c\,y = bc - by \end{cases} \Rightarrow c\,\frac{x}{\sqrt{2}} = bc - b\,\frac{x}{\sqrt{2}} \Rightarrow x = \frac{\sqrt{2}\,bc}{b + c}.$$

Num isosceles triangle $ABC$, $M$ is any point in the base $\overline{BC}$. Show that $(AB)^2 - (AM)^2 = (MB) \cdot (MC)$.

Determine the perimeter of a isosceles triangle as a function of the projection $a$ of the height relative to the base of the triangle on one of the congruent sides, and as a function of this height $h$.

In a square $ABCD$ we take a point $E$, on the side $\overline{AD}$, such as $AE = \dfrac{1}{4}AD$, and the point $O$, middle of $\overline{AB}$. Since $\overline{OP}$ is perpendicular to $\overline{CE}$, where $P$ is the foot of the perpendicular taken over $\overline{CE}$, prove that $(OP)^2 = (EP) \cdot (CP)$.

Let us consider a triangle $ABC$ and the angle bisectors $\overline{AD}$ internal and $\overline{AE}$ external to the triangle. Prove that

Let there be a semicircle of diameter $AB = 2r$ and the tangents $\overrightarrow{AX}$ and $\overrightarrow{BY}$ to the semicircle. The tangent at any point $C$ on the semicircle meets $\overrightarrow{AX}$ in $D$ and $\overrightarrow{BY}$ in $E$. Show that $(CD) \cdot (CE) = r^2$.

Determine $\operatorname{sen}\alpha$ in cases:

Determine $\cos\alpha$ in cases:

Get $\operatorname{tg}\alpha$ in cases:

Determine the value of $x$ in the following cases:

Determine the value of $x$ in the following cases:

Determine the values ​​of $x$ and $y$ in the cases: a) rectangle; b) parallelogram; c) parallelogram; d) rectangular trapezoid; e) isosceles trapezoid.

Determine the value of $x$ in the following cases:

A point on one side of an angle of 60° is 16 m from the vertex of the angle. How far is it on the other side of the angle?

A point on one side of an angle of 30° is 6 m away from the other side. Determine the distance from the orthogonal projection of this point on this other side to the vertex of the angle.

An internal point $P$ of a right angle is 4 m and 8 m from the sides of the angle. What is the distance between $P$ and the vertex of this angle?

An internal point of a right angle is 4 m and 10 m from the sides of the angle. How far is this point from angle bisector of this angle?

A point $P$, inside a right angle, is respectively $\sqrt{2}$ m and 2 m away from one side and angle bisector of the angle. Determine the distance between $P$ and the vertex of this angle.

A point $P$, inside an angle of 60°, is 6 m and 9 m away from the sides of that angle. What is the distance between $P$ and angle bisector of the angle?

A point $P$, inside an angle of 60°, is 3 m and 6 m away from the sides of the angle. Determine the distance between $P$ and the vertex of this angle.

A point $P$, inside an angle of 30°, is 3 m away from one side and $3\sqrt{13}$ m from the vertex of the angle. How far is this point on the other side of the angle?

A point $P$, external to an angle of 60°, is $9\sqrt{3}$ m and $3\sqrt{3}$ m away from the sides of the angle, none of these distances being to the vertex of the angle. What is the distance between $P$ and angle bisector of the angle?

In a right triangle, the square of the hypotenuse is twice the product of the legs. Calculate one of the acute angles of the triangle.

Through the vertex $B$ of a square $ABCD$ with side $a$, a segment $\overline{BS}$ is taken inside the square that forms an angle equal to 30° with $\overline{BA}$, with $S$ in $\overline{AD}$. Determine $AS$ and $BS$.

An observer views a building, built on level ground, at an angle of 60°. If he moves further away from the building 30 m, he will see it at an angle of 45°. Calculate the height of the building.

The sides $\overline{AB}$ and $\overline{AC}$ of a triangle $ABC$ measure respectively $a$ and $2a$, with 45° being the angle formed by them. Calculate the measurement of the height $\overline{BD}$ and the side $\overline{BC}$ of the triangle, as a function of $a$.

The bases of a rectangular trapezoid are $b$ and $2b$ and one of the angles measures 60°. Calculate the height.

One of the acute angles of an isosceles trapezoid measures 60°. Since the non-parallel sides are congruent to the smaller base of the trapezoid and $m$ is the measurement of the larger base, determine the perimeter of the trapezoid in terms of $m$.

Determine the angle that the diagonal of an isosceles trapezoid forms with the height of the trapezoid, knowing that the height of the trapezoid is equal to its average base multiplied by $\sqrt{3}$.

The largest base of an isosceles trapezoid measures 100 cm and the smallest base, 60 cm. With 60° being the measure of each of its acute angles, determine the height and perimeter of the trapezoid.

Determine $\operatorname{tg}\alpha$, knowing that $E$ is the midpoint of the side $\overline{BC}$ of the square $ABCD$.

Determine the radius of a circle inscribed in a circular sector of 60° and 6 dm in radius.

Let $AB = 3r$ be tangent in $A$ to a circle with center $O$ and radius $r$. Draw the tangent $\overline{BC}$ through $B$, which has $C$ as a contact point. Calculate the distance from $C$ to the straight line $\overleftrightarrow{AB}$.

Let us consider a right-angled triangle $ABC$, where the measure of an acute angle is $\alpha$. Determine the measurement of the radius of the inscribed circle as a function of $\alpha$ and the hypotenuse $a$.

A parallelogram has sides respectively equal to 10 cm and 8 cm. Knowing that one of its internal angles is worth 120°, calculate the perimeter of the convex quadrilateral formed by the angle bisectors of its internal angles.

In the figure we have a square and a equilateral triangle. Determine the unknowns.