Exercises and resolutions

Exercise 274 — Classify as true (V) or false (F)

  1. incenter is the center of the circle inscribed in the triangle.
  2. circumcenter is the center of the circle circumscribed by the triangle.
  3. incenter is internal to the triangle.
  4. centroid is internal to the triangle.
  5. orthocenter is internal to the triangle.
  6. circumcenter is internal to the triangle.
  7. centroid is the center of the circle inscribed in the triangle.
275

Say which triangle satisfies the given condition in the cases: a) orthocenter and centroid are coincident; b) incenter and circumcenter are coincident; c) orthocenter is one of the vertices; d) orthocenter is external; e) circumcenter is external; f) circumcenter is on one side; g) orthocenter is an internal point.

Ver resposta

a) equilateral · b) equilateral · c) rectangle · d) obtuse · e) obtuse · f) rectangle · g) acutangle

276

Consider the segments made up of the three heights, the three medians and the three internal angle bisectors of a triangle. How many of these segments, two by two distinct, will we have: a) in equilateral triangle; b) in isosceles triangle not equilateral; c) in scalene triangle?

Ver resposta

a) 3 · b) 7 · c) 9

277

With $G$ being centroid of the triangle $ABC$, determine $x$, $y$ and $z$, with $AG = 10$, $BG = y$ and $CG = 14$.

Triangle ABC with the medians and centroid G.
Ver resposta

$x = 7$, $y = 12$ e $z = 5$

278

If the quadrilateral $ABCD$ is a parallelogram and $M$ is the midpoint of $\overline{AB}$, determine $x$, with $DP = 16$ and $PM = x$.

Parallelogram ABCD with M midpoint of AB and P on the diagonal AC.
Ver resposta

$x = 8$: draw the diagonal $\overline{BD}$; $P$ is the centroid of the triangle $ABD$.

279

Being $H$ the orthocenter of a triangle $ABC$ and $B\hat{H}C = 150^{\circ}$, determine $\hat{A}$.

Ver resposta

$\hat{A} = 30^{\circ}$

280

If $H$ is the orthocenter of a isosceles triangle $ABC$ base $\overline{BC}$ and $B\hat{H}C = 50^{\circ}$, determine the angles of the triangle.

Ver resposta

$25^{\circ}$, $25^{\circ}$ e $130^{\circ}$

281

If $P$ is the incenter of a triangle $ABC$ and $B\hat{P}C = 125^{\circ}$, determine $\hat{A}$.

Ver resposta

$\hat{A} = 70^{\circ}$

282

The circumcenter of a isosceles triangle is internal to the triangle, and two perpendicular bisectors form an angle of $50^{\circ}$. Determine the angles of this triangle.

Ver resposta

($50^{\circ}$, $50^{\circ}$, $80^{\circ}$) or ($65^{\circ}$, $65^{\circ}$, $50^{\circ}$)

283

Considering the segments with “equal marks” to be congruent, determine the values ​​of the unknowns in the cases:

Triangle with medians and parallelogram with measurements 8 and x.
Ver resposta

a) $x = 4$ e $y = 6$ · b) $x = 4$

284

Considering the four notable points of a triangle: a) which ones can be external to the triangle? b) What could be the midpoint of one side? c) What could be the vertex of the triangle?

Ver resposta

a) o circumcenter e o orthocenter · b) o circumcenter · c) o orthocenter

285

In a triangle $ABC$, the angles $\hat{A}$ and $\hat{B}$ measure, respectively, $86^{\circ}$ and $34^{\circ}$. Determine the acute angle formed by perpendicular bisector relative to the side $\overline{BC}$ and by angle bisector of the angle $\hat{C}$.

Ver resposta

$60^{\circ}$

286

In a triangle $ABC$, the angles $\hat{A}$ and $\hat{B}$ measure, respectively, $70^{\circ}$ and $60^{\circ}$. Determine the ratio between the two largest angles formed by the intersections of the three heights.

Ver resposta

$\dfrac{13}{12}$ or $\dfrac{12}{13}$

287

Determine the measures of the three obtuse angles formed by perpendicular bisectors of one equilateral triangle.

Ver resposta

$120^{\circ}$ each

288

In the figure, $Q$ is the midpoint of $\overline{AB}$, and $\overline{QP}$ is parallel to $\overline{BC}$. Being $AC = 30$ cm, determine $PO$.

Right triangle ABC with segment QP parallel to BC and point O.
Ver resposta

$PO = 5$ cm

289

In the figure, $ABCD$ is a rectangle, $M$ is the midpoint of $\overline{CD}$, and the triangle $ABM$ is equilateral. Being $AB = 15$, calculate $AP$.

Rectangle ABCD with equilateral triangle ABM and point P.
Ver resposta

$AP = 10$; note that $P$ is the centroid of the triangle $ACD$.

290

Determine the perimeter of the triangle $ARS$ in the figure, where $AB$ and $AC$ measure 15 cm and 18 cm, respectively, with $\overline{BQ}$ and $\overline{CQ}$ being the angle bisectors of the angles $\hat{B}$ and $\hat{C}$ of the triangle $ABC$ and $\overline{RS}$ parallel to $\overline{BC}$.

Triangle ABC with angle bisectors BQ and CQ and segment RS parallel to BC.
Ver resposta

33 cm; note that triangles $RBQ$ and $SCQ$ are isosceles.

291

The three angle bisectors of a triangle $ABC$ meet at a point $O$. Determine the measures of the angles $A\hat{O}B$, $A\hat{O}C$ and $B\hat{O}C$ as a function of the angles $\hat{A}$, $\hat{B}$ and $\hat{C}$ of the triangle.

Ver resposta

$A\hat{O}B = 90^{\circ} + \dfrac{\hat{C}}{2}$; $A\hat{O}C = 90^{\circ} + \dfrac{\hat{B}}{2}$; $B\hat{O}C = 90^{\circ} + \dfrac{\hat{A}}{2}$