Exercises from chapter 15 — Any triangles

Knowing that $\operatorname{sen}(180^{\circ} - x) = \operatorname{sen}x$ and $\cos(180^{\circ} - x) = -\cos x$, determine: a) $\operatorname{sen}120^{\circ}$; b) $\cos 150^{\circ}$; c) $\cos 135^{\circ}$; d) $\operatorname{sen}150^{\circ}$; e) $\operatorname{sen}135^{\circ}$; f) $\cos 120^{\circ}$.

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

Determine the radius of the circle circumscribed by the triangle in the following cases:

Obtain the value of $x$ in the cases: a) $ABCD$ is a parallelogram; b) $ABCD$ is an isosceles trapezoid.

Determine the angle $x$ in the cases:

In a triangle $ABC$ $\hat A = 60^{\circ}$, $\hat B = 45^{\circ}$ and $BC = 4$ cm are given. Determine the measurement of $\overline{AC}$.

In an obtuse angled and isosceles triangle, the base angles measure 30° each. Determine the base of the triangle, knowing that the congruent sides measure 10 cm each.

The triangle $ABC$ is obtuse with $\hat A = 120^{\circ}$, $BC = 2\sqrt{3}$ dm and $AC = 2$ dm. Determine the measure of the vertex angle $B$ of this triangle.

If the sides of a triangle $ABC$ measure $a$, $b$ and $c$, prove that

where $A$ and $B$ are the angles of the vertices $A$ and $B$ of the triangle and $a$, $b$ and $c$ the sides respectively opposite the vertices $A$, $B$ and $C$.

In the same previous exercise, prove that $\dfrac{a - b}{b} = \dfrac{\operatorname{sen}A - \operatorname{sen}B}{\operatorname{sen}B}$.

Let $a$, $b$ and $c$ be the sides opposite the vertices $A$, $B$ and $C$ of a triangle $ABC$. Prove that $\dfrac{\operatorname{sen}A + \operatorname{sen}B}{\operatorname{sen}C} = \dfrac{a + b}{c}$.

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

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

Calculate the height $h$, relative to the side $\overline{BC}$, in the following cases:

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

Determine the measure $x$ of the angle in the following cases:

If the diagonals of the parallelogram in the figure measure 20 cm and 32 cm and form an angle of 60°, determine the sides of the parallelogram.

Recognize the nature of a triangle: a) whose sides measure 6, 12 and 13; b) whose sides measure 6, 10 and 12; c) whose sides measure 5, 12 and 13; d) whose sides are in the ratio 3 : 4 : 4,5; e) whose sides are inversely proportional to 3, 4 and 6.

Recognize the nature of a triangle whose sides are inversely proportional to the numbers 3, 4 and 5.

The sides of a triangle measure 15 m, 20 m and 25 m. Determine the height relative to the longest side.

The sides of a triangle measure 12 m, 20 m and 28 m. Determine the projection of the smallest onto the straight line on the side of 20 m.

The sides of a triangle measure 7 m, 24 m and 25 m. Determine the height relative to the short side.

Determine the side $\overline{BC}$ of an acute triangle $ABC$, where $AC = 7$ cm, $AB = 5$ cm and the projection of $\overline{AC}$ on $\overline{AB}$ measures 1 cm.

Determine the measurement of the side $\overline{AB}$ of a triangle $ABC$, obtuse angled at $A$, with $BC = 8$ cm, $AC = 5$ cm and the projection of the side $\overline{AB}$ on $\overline{AC}$ equal to 3 mm.

Determine the measurement of the side $\overline{BC}$ of a triangle $ABC$, where $AC = 10$ cm, $AB = 6$ cm and the projection of the side $\overline{BC}$ on $\overline{AC}$ is worth 10,4 cm.

The base of a triangle measures 10 cm and the other two sides, 14 cm and 8 cm, respectively. Determine the projections of these two sides onto the base of the triangle.

Determine the projection of side $\overline{BC}$ onto side $\overline{AC}$ of a triangle $ABC$, where $BC = 12$ cm, $AC = 16$ cm and $AB = 18$ cm.

In a triangle $ABC$ is it possible to have $a^2 = b^2 + c^2 + 2bm$ and $c^2 = b^2 + a^2 + 2bn$ simultaneously, with $m$ being the projection of $c$ onto $b$ and $n$ being the projection of $a$ onto $b$? Justify.

In the figure, calculate the value of $x$.

In the triangle $ABC$ in the figure, the side $\overline{AB}$ measures 12 cm, the side $\overline{BC}$ measures 9 cm and the side $\overline{AC}$ measures 6 cm. Calculate the cosine of the angle $\alpha$.

Calculate the perimeter of the triangle in the figure.

Calculate the perimeter of the triangle $ABC$ in the figure.

Determine the value of $x$, knowing that $x + 5$, $3 - x$ and $x + 7$ are the measurements of the sides $\overline{AB}$, $\overline{BC}$ and $\overline{AC}$ of a triangle $ABC$ whose angle $\hat B$ is equal to 120°.

A chord $\overline{AB}$ of measurement $\ell$ determines an arc of 120° on a circle. Determine the distance from point $B$ to diameter $\overline{AC}$ of this circle.

In the figure, $\overline{AB}$ is equal to the radius of the center circle $O$, $BC = 26$ and $\overline{BH}$ is perpendicular to $\overline{AC}$. Calculate $HC$.

Determine the longest diagonal of a parallelogram, in which two of its consecutive sides form an angle of 45° and measure, respectively, $5\sqrt{2}$ cm and 10 cm.

Knowing that the consecutive sides of a parallelogram measure 4 cm and 5 cm and one of the diagonals measures 6 cm, determine the measurement of the other diagonal.

Two consecutive sides of a parallelogram measure 8 m and 12 m and form an angle of 60°. Calculate the diagonals.

The median $\overline{AM}$ of a triangle $ABC$ measures 6 cm, divides the opposite side into two segments equal to 12 cm and forms two angles with this side that differ by 60°. Determine the lengths of the sides of this triangle.

Is there a triangle $ABC$ such that $BC = 10$ cm, $AC = 1$ cm and $\beta = 30^{\circ}$, where $\beta$ is the angle opposite the side $\overline{AC}$?

Two consecutive sides of a parallelogram have measurements $a$ and $b$ and one of the diagonals has dimensions $c$. Determine the length of the other diagonal.

Prove that: “In any triangle, the square on the side opposite an acute angle is equal to the sum of the squares on the other two sides, minus twice the product of one of these sides and the projection of the other onto it.”

Prove that: “In an obtuse triangle, the square on the side opposite the obtuse angle is equal to the sum of the squares of the other two, plus twice the product of one of these sides and the projection of the other onto it.”

If, in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus the product of these two sides, calculate the internal angle that the same two sides form.

On the sides of a triangle $ABC$ rectangle with sides 6 cm, $6\sqrt{3}$ cm and 12 cm we construct three external squares. Calculate the length of the sides of the triangle determined by the centers of these squares.

The measurements of the sides of the quadrilateral $ABCD$ are $AB = BC = 10$ m, $CD = 16$ m and $AD = 6$ m. Determine $BD$.

An internal point of equilateral triangle is 5 cm, 7 cm and 8 cm away from the vertices of the triangle. Determine the side of this triangle.

See the resolution of the book

1. Consider a point $Q$ external to the triangle with $BQ = 5$ and $CQ = 7$. Note that $\triangle PBA \equiv \triangle QBC$ (LLL), which gives $P\hat BQ = 60^{\circ}$. So $\triangle PBQ$ is equilateral and $B\hat PQ = 60^{\circ}$ ($\alpha = 60^{\circ}$).
2. Apply the law of cosines to $\triangle PQC$ (sides 5, 7 and 8): we obtain $\beta = 60^{\circ}$.
3. With $\alpha = 60^{\circ}$ and $\beta = 60^{\circ}$, $B\hat PC = 120^{\circ}$. Applying the law of cosines to $\triangle BPC$: $x^2 = 5^2 + 8^2 - 2 \cdot 5 \cdot 8 \cos 120^{\circ} = 129$, that is, $x = \sqrt{129}$ cm.

Determine the measurement of the median $\overline{AM}$ of the triangle $ABC$, applying the median formula, and then calculate using the Stewart relationship.

Determine the measure of angle bisector $\overline{AS}$, applying the internal angle bisector formula, and then calculate using the angle bisector theorem and the Stewart relation.

Determine the measurement of angle bisector outside $\overline{AP}$ of $\triangle ABC$, applying the angle bisector formula, and then calculate using the angle bisector theorem and Stewart’s relation.

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

Determine the ratio between the sum of the squares of the medians of a triangle and the sum of the squares of the sides of that triangle.

Calculate the heights of a triangle whose sides measure 6 m, 10 m and 12 m.

The sides $\overline{AB}$, $\overline{AC}$ and $\overline{BC}$ of a triangle $ABC$ measure, respectively, 5 cm, 6 cm and 7 cm. Determine the height and the internal angle bisector relative to the $\overline{AC}$ side and the external angle bisector relative to the $\overline{AB}$ side.

Given the sides $a$, $b$ and $c$ of a triangle $ABC$, calculate the distance from the vertex $A$ to the point $M$ that divides the base $\overline{BC}$ into segments equal to $m$ and $n$.

If $\overline{AS}$ is angle bisector inside the triangle $ABC$, determine $x$ and $y$.

If $\overline{AP}$ is angle bisector external to the triangle $ABC$, determine $x$ and $y$.

Derive the formulas that give the three medians $m_a$, $m_b$ and $m_c$ of a triangle, depending on the sides $a$, $b$ and $c$.

Derive the formulas that give the three heights $h_a$, $h_b$ and $h_c$ of a triangle as a function of the sides $a$, $b$ and $c$.

Derive the expressions that provide the internal angle bisectors $s_a$, $s_b$ and $s_c$ of a triangle in function of the sides $a$, $b$ and $c$.

Given the three sides $a$, $b$, and $c$ of a triangle, obtain $s^{\prime}_a$, $s^{\prime}_b$, and $s^{\prime}_c$ by deducing the formulas that give the outer angle bisectors.