Tests — Triangles

Among the right-angled triangles below, one and only one presents the correct data. Mark it:

Considering the triangle $ABC$, a rectangle in $A$, with $a = 7{,}5$ m, $b = 4{,}5$ m and $c = 6$ m, with $x$ being the angle at the vertex $B$, it can be stated that $\operatorname{tg} x$ is equal to:

In the triangle $ABC$, a rectangle in $A$, $BC = 10$ and $\cos\alpha = \frac{3}{5}$ are given, with $\alpha$ being the angle in $B$. The sum of the legs is:

In the figure, the rectangle has height 50; the straight line that goes from the bottom left corner to $A$ forms $60^\circ$ with the base, and the one that goes from the top left corner to $A$ forms $30^\circ$ with the top side. $A$ is vertical to $B$. $AB$ is worth:

$\overline{AB}$ is the hypotenuse of a right triangle $ABC$. The median $\overline{AD}$ measures 7 and the median $\overline{BE}$ measures 4. The length of $\overline{AB}$ is equal to:

In the triangle $ABC$ we know that $\hat{A} = 90^\circ$, $\hat{B} = 60^\circ$ and $AB = 50$ cm. Then segment $\overline{AC}$ measures:

A person 1,70 m tall observes the top of a tree at an angle $\alpha$ with the horizontal; $b$ is the horizontal distance from the eyes to the trunk and $a$ is the distance from the eyes to the top. To know approximately the height of the tree, you must add 1,70 m with:

In the figure, $ABC$ is rectangle in $B$, $D$ is between $A$ and $B$, $AD = d$, $BC = h$, $C\hat{A}D = \alpha$ and $C\hat{D}B = \beta$. Then:

In the triangle $ABC$ there are $b = 2$, $\hat{B} = 45^\circ$ and $\hat{C} = 60^\circ$. Then the $a$ side measures:

In a right-angled triangle in which one side is worth 1 and the other is worth $\operatorname{tg}\varphi$, the hypotenuse is worth:

Consider the triangle $ABC$, a rectangle in $A$, with hypotenuse $a$ and legs $b = AC$ and $c = AB$, and indicate its area by $S$. Select the true statement:

In the figure, $c$ is a circle of radius $R$, $r$ and $s$ are tangent lines to the circle that intersect at $O$, $T$ is the point of tangency at $s$ and $OT = 2R$. Then the angle $\alpha$ of the straight lines $r$ and $s$ checks:

In a circle with center $O$ and radius $R$, a measuring chord $\frac{3R}{2}$ determines the central angle $\alpha$. Regarding $\alpha$, it can be said that:

In a triangle $ABC$ it is assumed that $2\operatorname{tg}\hat{A} = \operatorname{tg}\hat{B} + \operatorname{tg}\hat{C}$ and $0 < \hat{A} < \frac{\pi}{2}$. In this triangle the relationship applies:

Consider a triangle $ABC$ whose internal angles $\hat{A}$, $\hat{B}$ and $\hat{C}$ verify the relationship $\operatorname{sen}\hat{A} = \operatorname{tg}\frac{\hat{B} + \hat{C}}{2}$. So we can say that:

In a triangle $ABC$, the angles $\hat{A}$ and $\hat{B}$ measure, respectively, $60^\circ$ and $45^\circ$; the side $\overline{BC}$ measures $5\sqrt{6}$ cm. So the measurement of side $\overline{AC}$ is:

A ship, sailing in a straight line, passes successively through the points $A$, $B$, $C$. When the ship is at $A$, the captain observes a beacon $L$ and calculates the angle $L\hat{A}C = 30^\circ$. After navigating 4 miles to $B$, check the $L\hat{B}C = 75^\circ$ angle. How many miles separate the lighthouse from point $B$?

In the figure, $\overline{AC} \perp \overline{CB}$, $\overline{AD} \perp \overline{DC}$, $m(D\hat{A}C) = m(A\hat{B}C) = \beta$ and $AB = a$. The value of $AD$, as a function of $a$ and $\beta$, is:

It is desired to build a railway connecting point $A$ to point $B$, which is $40\sqrt{2}$ km southeast of $A$. A lake, on the plain where $A$ and $B$ are, prevents construction in a straight line. To bypass the lake, the road will be built in 2 straight sections with apex at point $C$, which is 36 km east and 27 km south of $A$. The length of the section $\overline{CB}$ is:

The sides of a triangle are in the ratio $6 : 8 : 9$. Then:

Given the segments $\overline{AC}$ and $\overline{BC}$ and an angle $\hat{B}$, it is possible to construct a triangle that has $\overline{AC}$ and $\overline{BC}$ as sides and $\hat{B}$ as an angle not adjacent to $\overline{AC}$ when:

In a scalene triangle $ABC$, the sides opposite the angles $\hat{A}$, $\hat{B}$, $\hat{C}$ measure, respectively, $a$, $b$, $c$. Then the expression $a\operatorname{sen}(\hat{B} - \hat{C}) + b\operatorname{sen}(\hat{C} - \hat{A}) + c\operatorname{sen}(\hat{A} - \hat{B})$ has a value that satisfies one of the following alternatives:

The side of the regular octagon inscribed in a circle of unit radius is $\sqrt{2 - \sqrt{2}}$. It can be concluded that $\cos\frac{\pi}{8}$ is worth:

The base $\overline{AD}$ of a rectangle, which is three times greater than its height $\overline{AB}$, is subdivided by the points $M$ and $N$ into three parts of equal size. Under these conditions, $A\hat{M}B + A\hat{N}B + A\hat{D}B$ is equal to:

Let $ABCD$ be a convex quadrilateral inscribed in a circle. It is known that $\hat{A} = 2\hat{C}$, $\hat{B} > \hat{D}$ and $\operatorname{tg}\hat{B}\operatorname{tg}\hat{D} + \operatorname{sen}\hat{A}\operatorname{sen}\hat{C} = -\frac{9}{4}$. In this case, the values ​​of $\hat{A}$, $\hat{B}$, $\hat{C}$, $\hat{D}$ are, respectively:

Let $A$, $B$ and $C$ be three distinct points on a straight line, with $B$ between $A$ and $C$. Let $a$ and $b$ ($a > 2b$) be the lengths of $\overline{AB}$ and $\overline{BC}$, respectively. If the segment $\overline{BD}$ is perpendicular to the segment $\overline{AC}$, how much must $BD = x$ measure so that the angle $B\hat{D}C$ is half of $B\hat{D}A$?

Let $d$ and $L$, respectively, be the lengths of the diagonal $\overline{BD}$ and the side $\overline{BC}$ of the parallelogram $ABCD$. The diagonal $\overline{BD}$ forms the angle $\alpha$ with $\overline{BC}$ and the angle $\beta$ with $\overline{BA}$. The length $x$ of side $\overline{AB}$ is given by: