Exercises from chapter IX — Right triangles

Calculate the elements $a$, $h$, $m$ and $n$ indicated in the figure (categories $b = 4$ and $c = 3$).

Calculate the elements $x$, $y$, $z$, $t$ in the figure: the legs measure 12 and 5, $t$ is the hypotenuse, $x$ is the height relative to it and $y$ and $z$ are the projections of the legs 5 and 12 on the hypotenuse.

Write 6 metric relations involving the elements of the figure: a right-angled triangle with legs $\alpha$ and $\beta$ and hypotenuse $\gamma$, with height $\varphi$ relative to the hypotenuse, which is divided into projections $\delta$ (from $\alpha$) and $\varepsilon$ (from $\beta$).

Calculate the area of ​​a right-angled triangle in which the projections of the legs on the hypotenuse measure, respectively, 4 and 9.

Calculate the hypotenuse of a right-angled triangle with perimeter 56 and height relative to the hypotenuse $\dfrac{168}{25}$.

A isosceles triangle $ABC$ has a base $a = 12$ and is inscribed in a circle of diameter $2R = 20$. Calculate the measurements of the sides $b$ and $c$ of the triangle.

Calculate the height of a isosceles triangle knowing the radius $R$ of the circumscribed circle and the base $a$. Data: $R = 5$ and $a = 8$.

Calculate the radius of the circle circumscribed by a isosceles triangle base 6 whose other sides measure $\sqrt{90}$.

A point $P$ is distant $d = 13$ from the center $O$ of a circle of radius $R = 5$. If we draw through $P$ a tangent line to the circle at point $T$, what is the measurement of the segment $\overline{PT}$?

Calculate the side of a regular octagon inscribed in a circle of radius $R$.

Calculate the internal angles of a right-angled triangle whose legs are $b = 9$ and $c = 12$.

Calculate the sides of a right triangle knowing that the height relative to the hypotenuse is $h = 4$ and an acute angle is $\hat{B} = 30^\circ$.

Calculate the sides of a right-angled triangle knowing that the height relative to the hypotenuse measures 4 and forms an angle of $15^\circ$ with the side $b$.

Calculate the legs of a right-angled triangle $ABC$ knowing that it is inscribed in a circle of radius 3 and has acute angles such as $\hat{B} = 2\hat{C}$.

Calculate the acute angles of a right-angled triangle with hypotenuse 20, knowing that the median relative to one of the legs measures 15.

(FFCLUSP-66) In the figure, the angles $O\hat{A}_iB_i$ and $O\hat{B}_{i+1}A_i$ ($i = 0, 1, 2, 3, \ldots$) are right angles. How much is the sum of the segments $\overline{A_0B_0}$, $\overline{A_1B_1}$, $\overline{A_2B_2}$, … worth in terms of $\overline{A_0B_0}$ and $\theta$?

Calculate the angle formed by the diagonal and the smallest side of a rectangle whose sides are in the ratio $\dfrac{3}{4}$.

Calculate the area of ​​a isosceles triangle $ABC$ whose base is $a = 8$, knowing that $\hat{A} = 45^\circ$.

Calculate the height relative to the base of a isosceles triangle with perimeter $2p = 36$, knowing that the angles adjacent to the base are equal to $\operatorname{arc\,cos}\dfrac{2}{7}$.

Calculate the radius of the circle circumscribed to a isosceles triangle base $a = 8$ and angle opposite the base $\hat{A} = 120^\circ$.

A straight line determines, on a circle of radius 10, a chord of length 16. What is the measurement of the central angle from which the rope is seen?

Determine the angle $\hat{B}$ of a right triangle $ABC$ in $A$ knowing that the relationship $\dfrac{1}{c} + \dfrac{2}{b} = \dfrac{\sqrt{5}}{h}$ is verified, where $h$ is the height relative to the hypotenuse.

In a triangle $ABC$ right angled to $A$, it is known that the acute angle formed by the medians $\overline{BM}$ and $\overline{CN}$ is $\theta = 30^\circ$. Calculate the angle $\hat{C}$.

In a right triangle $ABC$ in $A$ the internal angle bisectors $\overline{BB'}$ and $\overline{CC'}$ are drawn. Knowing that $AB' = 1$ and $AC' = 1$, calculate the angle $\hat{B}$ and the hypotenuse $a$.

Calculate the acute angles of the right triangle with hypotenuse $a = 13$ m, knowing that the radius of the inscribed circle is $r = 2$ m.

An observer sees a building, built on level ground, at an angle of $60^\circ$. Moving further away from the building 30 m, you can see it from an angle of $45^\circ$. How tall is the building?

(MAUÁ-68) To measure the height of the vertical tower $\overline{DE}$, the segment $\overline{AB}$ of length 12 m and whose midpoint is $C$ is taken in the horizontal plane that passes through its base $D$. The angles $D\hat{A}E$, $D\hat{B}E$ and $D\hat{C}E$ are measured, verifying that $D\hat{A}E = D\hat{B}E = 45^\circ$ and $D\hat{C}E = 60^\circ$. Determine the height of the tower.

Calculate the distance between the sills of two windows of a skyscraper, knowing the angles $\alpha$ and $\beta$ ($\alpha < \beta$) under which they are observed from a point $O$ on the ground and the distance $d$ from that point to the building.

(MAUÁ-65) To obtain the height $H$ of a chimney $\overline{CD}$, an engineer, with a special device, established the horizontal $\overline{AB}$ ($B$ in the chimney) and measured the angles $\alpha = B\hat{A}C$ (down, to the base $C$) and $\beta = B\hat{A}D$ (up, to the top $D$), having the follow measurement $BC = h$. Determine the height of the chimney.

(MAUÁ-67) An observer is on a highway, in a straight and horizontal section located on the same vertical plane that contains a TV tower at the top of a peak. From two positions $A$ and $B$ of this section, distant 60 m from each other, the observer sees the upper end of the tower, respectively, under the angles of $30^\circ$ and $31^\circ 53'$. The device that measures the angles was placed 1,50 m above the track, which is 721,50 m above sea level. Determine the height of the end of the tower in relation to sea level. Data: $\operatorname{tg} 31^\circ 53' \approx 0{,}62$.

(LINS-66) In the figure, $A$, $C$ and $B$ are aligned on the floor with $AC = CB = 60$ m; $\overline{AM} = x$ and $\overline{BN} = y$ are vertical; $N\hat{A}B = 2\alpha$, $M\hat{B}A = \alpha$ and $M\hat{C}N = 90^\circ$. Calculate the distances $x$ and $y$.

Solve a right triangle $ABC$ knowing the measure of the internal angle bisector $S_b = 5$ and the angle $\hat{C} = 30^\circ$.

Solve a right triangle $ABC$ given $b = 3$ and $a - c = \sqrt{3}$.

Solve a right triangle $ABC$ in $A$ knowing that $a + b = 18$ and $a + c = 25$.

Solve a right triangle $ABC$ knowing that $a = 4$ and that the measure of angle bisector inside $\overline{BE}$ is $S_b = 6\sqrt{2} - 2\sqrt{6}$.

Solve a right triangle knowing the height $h = 1$ relative to the hypotenuse and the perimeter $2p = 2\sqrt{2} + 2$.

Solve a isosceles triangle $ABC$ knowing that the height relative to the base $\overline{BC}$ measures $h = 24$ and the perimeter is $2p = 64$.

Solve a right triangle $ABC$ knowing the radius $r = 2$ of the inscribed circle and the height $h = \dfrac{60}{13}$ relative to the hypotenuse.

Solve a right-angled triangle $ABC$ knowing the height $h = 2$ relative to the hypotenuse and the radius $r' = 2\sqrt{2} + 2$ of the ex-inscribed circle located at the right angle.