Euler's Eyes
2026-10-11
Chapter exercises 2 — Line segment
Exercises and resolutions
If the segment $\overline{AB}$ measures 17 cm, determine the value of $x$ in the cases:
Ver resolução
a) $x + 7 = 17 \Rightarrow x = 10$. b) $PA = PB + BA$: $21 = x + 17 \Rightarrow x = 4$. c) $(x + 3) + x = 17 \Rightarrow x = 7$. d) $AP = AB + BP$: $2x = 17 + (x - 3) \Rightarrow x = 14$.
Ver resposta
a) $x = 10$ cm · b) $x = 4$ cm · c) $x = 7$ cm · d) $x = 14$ cm
Determine $x$, with $M$ the midpoint of $\overline{AB}$:
Ver resposta
a) $x = 7$ · b) $x = 6$
Determine $PQ$, where $AB = 31$:
Ver resolução
a) $AB = AQ + QB = 2x + (x + 1) = 31 \Rightarrow x = 10$. Then $AP = 9$, $AQ = 20$ and $PQ = 20 - 9 = 11$.
b) $AB = AP + PB = x + 11 = 31 \Rightarrow x = 20$. Then $PQ = PB + BQ = 11 + 21 = 32$.
Ver resposta
a) $PQ = 11$ · b) $PQ = 32$
Determine $AB$, with $M$ the midpoint of $\overline{AB}$:
Ver resolução
a) $2x - 5 = x + 8 \Rightarrow x = 13$, then $AB = 2 \cdot 21 = 42$.
b) $AM = MB = x$ and $AP = AB + BP$: $4x - 5 = 2x + (x + 7) \Rightarrow x = 12$, then $AB = 24$.
Ver resposta
a) $AB = 42$ · b) $AB = 24$
How many rays are there in a straight line, originating at the four points $A$, $B$, $C$ and $D$ on the straight line?
Ver resposta
8 rays: from each of the 4 points, 2 opposite rays depart.
Three distinct points on a straight line, how many distinct segments can you determine?
Ver resposta
3 segments: $\overline{AB}$, $\overline{AC}$ and $\overline{BC}$.
How many segments are there that pass through different points $A$ and $B$? How many are there with ends $A$ and $B$?
Ver resposta
Infinitely many segments pass through $A$ and $B$; with ends $A$ and $B$ there is a single one.
Exercise 13 — Classify as true (V) or false (F)
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If two segments are consecutive, then they are collinear.
Falso. $\overline{AB}$ and $\overline{BC}$ can form a “beak”, outside the same straight line.
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If two segments are collinear, then they are consecutive.
Falso. They can be on the same line with no common end.
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If two segments are adjacent, then they are collinear.
Verdadeiro. The definition of adjacent requires that they be collinear.
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If two segments are collinear, then they are adjacent.
Falso. They would need to be consecutive without common internal points.
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If two segments are adjacent, then they are consecutive.
Verdadeiro. It is also part of the definition.
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If two segments are consecutive, then they are adjacent.
Falso. $\overline{RS}$ and $\overline{ST}$ overlapping are consecutive and not adjacent.
The segment $\overline{AB}$ of a straight line is equal to five times the segment $\overline{CD}$ of that same straight line. Determine the measurement of $\overline{AB}$, considering the fifth part of the segment $\overline{CD}$ as the unit of measurement.
Ver resposta
$AB = 25$ units.
$P$, $A$ and $B$ are three distinct points on a straight line. If $P$ is between $A$ and $B$, what relationship should be valid between the segments $\overline{PA}$, $\overline{PB}$ and $\overline{AB}$?
Ver resposta
$\overline{PA} + \overline{PB} = \overline{AB}$
$P$, $Q$ and $R$ are three distinct points on a straight line. If $\overline{PQ}$ is equal to three times $\overline{QR}$ and $PR = 32$ cm, determine the measurements of the segments $\overline{PQ}$ and $\overline{QR}$.
See the resolution of the book
Doing $QR = x$, we have $PQ = 3x$. There are two possibilities:
1st case: $3x + x = 32 \Rightarrow x = 8$, then $PQ = 24$ cm and $QR = 8$ cm.
2nd case: $3x = 32 + x \Rightarrow x = 16$, therefore $PQ = 48$ cm and $QR = 16$ cm.
Ver resposta
$PQ = 24$ cm and $QR = 8$ cm, or $PQ = 48$ cm and $QR = 16$ cm.
The segments $\overline{AB}$ and $\overline{BC}$, $\overline{BC}$ and $\overline{CD}$ are adjacent, in such a way that $\overline{AB}$ is triple $\overline{BC}$, $\overline{BC}$ is twice $\overline{CD}$ and $AD = 36$ cm. Determine the measurements of the segments $\overline{AB}$, $\overline{BC}$ and $\overline{CD}$.
Ver resposta
$AB = 24$ cm, $BC = 8$ cm and $CD = 4$ cm.
Let $P$, $A$, $Q$ and $B$ be points arranged on a straight line $r$, in that order. If $\overline{PA}$ and $\overline{QB}$ are congruent segments, show that $\overline{PQ}$ and $\overline{AB}$ are congruent.
Ver resposta
Outputs by sum: $\overline{PQ} = \overline{PA} + \overline{AQ}$ and $\overline{AB} = \overline{AQ} + \overline{QB}$; as $\overline{PA} \equiv \overline{QB}$, follows $\overline{PQ} \equiv \overline{AB}$.
If $A$, $B$ and $C$ are collinear points, determine $AC$, where $AB = 20$ cm and $BC = 12$ cm.
Ver resposta
$AC = 8$ cm or $AC = 32$ cm.
$\overline{AB}$ and $\overline{BC}$ are two adjacent segments. If $\overline{AB}$ is the quintuple of $\overline{BC}$ and $AC = 42$ cm, determine $AB$ and $BC$.
Ver resposta
$AB = 35$ cm and $BC = 7$ cm.
Being $\overline{AB}$ and $\overline{BC}$ consecutive collinear segments, $\overline{AB}$ the quadruple of $\overline{BC}$ and $AC = 45$ cm, determine $AB$ and $BC$.
Tip: Consecutive and collinear do not have to be adjacent — $C$ can be between $A$ and $B$.
Ver resposta
($AB = 36$ cm and $BC = 9$ cm) or ($AB = 60$ cm and $BC = 15$ cm).
On a straight line $r$, let us take the segments $\overline{AB}$ and $\overline{BC}$ and a point $P$ so that $\overline{AB}$ is the quintuple of $\overline{PC}$, $\overline{BC}$ is the quadruple of $\overline{PC}$ and $AP = 80$ cm. With $M$ and $N$ being the midpoints of $\overline{AB}$ and $\overline{BC}$, respectively, determine $MN$.
Ver resposta
$MN = 36$ cm, $45$ cm or $20$ cm, depending on the position of the points.
Let there be four points $A$, $B$, $C$, $D$ arranged on the same straight line $r$, in that order, and such that $\overline{AB}$ and $\overline{CD}$ are congruent. Show that the segments $\overline{AD}$ and $\overline{BC}$ have the same midpoint.
Ver resposta
$(\overline{BM} \equiv \overline{MC},\ \overline{AB} + \overline{BM} = \overline{AM},\ \overline{MC} + \overline{CD} = \overline{MD},\ \overline{AB} \equiv \overline{CD}) \Rightarrow \overline{AM} \equiv \overline{MD}$, where $M$ is the midpoint of $\overline{BC}$.
Let there be four points $A$, $B$, $C$, $D$ arranged on the same straight line, in that order, and such that the segments $\overline{AC}$ and $\overline{BD}$ are congruent. Show that the segments $\overline{AB}$ and $\overline{CD}$ are congruent and that the segments $\overline{BC}$ and $\overline{AD}$ have the same midpoint.
Ver resposta
$(\overline{AB} = \overline{AC} - \overline{BC},\ \overline{CD} = \overline{BD} - \overline{BC},\ \overline{AC} \equiv \overline{BD}) \Rightarrow \overline{AB} \equiv \overline{CD}$. Then, apply exercise 23.
Let $M$ and $N$ be the midpoints, respectively, of the segments $\overline{AB}$ and $\overline{BC}$, contained in the same straight line, being $\overline{AB} \equiv \overline{BC}$, with $A \neq C$. Show that $\overline{MN}$ is congruent to $\overline{AB}$.
Ver resposta
With $AB = BC = 2a$ and $B$ between $A$ and $C$: $MN = MB + BN = a + a = 2a = AB$.
The next three exercises can be explored in the following figure. Choose the exercise and drag the points: the relationship continues to apply in all positions.
Given three points $A$, $B$, $C$ on the same straight line, let us consider $M$ and $N$ the midpoints of the segments $\overline{AB}$ and $\overline{BC}$. Show that $\overline{MN}$ is equal to the semisum or semidifference of the segments $\overline{AB}$ and $\overline{BC}$.
Ver resposta
If $B$ is between $A$ and $C$: $MN = \dfrac{AB + BC}{2}$ (semisum). Otherwise: $MN = \dfrac{|AB - BC|}{2}$ (semi-difference).
Let $\overline{AB}$ be a straight line segment and $M$ its midpoint. Let us consider a point $P$ between the points $M$ and $B$. Show that $PM$ is given by the positive half-difference between $PA$ and $PB$.
See the resolution of the book
Indicating the measurement of $\overline{AB}$ by $2a$ and that of $\overline{PM}$ by $x$, we have $AM = MB = a$ and
$$PA = a + x \quad\text{e}\quad PB = a - x \ \Longrightarrow\ PA - PB = 2x \ \Longrightarrow\ PM = x = \frac{PA - PB}{2}.$$Let us consider on a straight line $r$ a fixed segment $\overline{AB}$ and a moving point $P$. Let $M$ be the midpoint of $\overline{AP}$ and $N$ be the midpoint of $\overline{BP}$. What can we say about segment $\overline{MN}$?
Ver resposta
$\overline{MN}$ has a constant measurement: $MN = \dfrac{AB}{2}$, whatever the position of $P$.