Exercises on proportional segments

When the ratio of two line segments is equal to the ratio of two other segments, they are called proportional segments.

A reason between two segments is obtained by dividing the length of one by the other.

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Thus, given four proportional line segments with lengths The, B, w It is d, in that order, we have a proportion:

\dpi{120} \mathbf{\frac{a}{b} \frac{c}{d}}

And, by the fundamental property of proportions, we have \dpi{120} \mathbf{ ad cb}.

To learn more, check out a list of exercises on proportional segments, with all questions resolved!

Exercises on proportional segments


Question 1. The segments \dpi{120} \overline{AB}, \overline{CD}, \overline{EF}\, \mathrm{e}\, \overline{GH} are, in that order, proportional segments. Determine the measure of \dpi{120} \overline{CD} knowing that \dpi{120} \overline{AB} 5, \dpi{120} \overline{EF} 7.5 It is \dpi{120} \overline{GH} 13.8.


Question 2. Determine \dpi{120} \overline{BC} knowing that \dpi{120} \frac{\overline{AB}}{7} \frac{\overline{BC}}{4} is that:

line segment

Question 3. Determine \dpi{120} \overline{AB} knowing that \dpi{120} \frac{\overline{AB}}{2} \frac{\overline{BC}}{5} is that:

line segment

Question 4. Determine the lengths of the sides of a triangle that has a perimeter of 52 units and whose sides are proportional to the sides of another triangle with lengths 2, 6, and 5.


Resolution of question 1

If the segments \dpi{120} \overline{AB}, \overline{CD}, \overline{EF}\, \mathrm{e}\, \overline{GH} are, in that order, proportional segments, then:

\dpi{120} \frac{\overline{AB}}{\overline{CD}} \frac{\overline{EF}}{\overline{GH}}

replacing \dpi{120} \overline{AB} 5, \dpi{120} \overline{EF} 7.5 It is \dpi{120} \overline{GH} 13.8, We have to:

\dpi{120} \frac{5}{\overline{CD}} \frac{7,5}{13,8}

Applying the fundamental property of proportions:

\dpi{120} \Rightarrow 7.5 \cdot \overline{CD} 69
\dpi{120} \Rightarrow \overline{CD} \frac{69}{7.5}
\dpi{120} \Rightarrow \overline{CD} 9.2

Resolution of question 2

We have:

\dpi{120} \frac{\overline{AB}}{7} \frac{\overline{BC}}{4}

replacing \dpi{120} \overline{AB} 11, We have to:

\dpi{120} \frac{11}{7} \frac{\overline{BC}}{4}

Applying the fundamental property of proportions:

\dpi{120} \Rightarrow 7\overline{BC} 44
\dpi{120} \Rightarrow \overline{BC} \frac{44}{7}
\dpi{120} \Rightarrow \overline{BC} \approx 6.28

Resolution of question 3

We have:

\dpi{120} \frac{\overline{AB}}{2} \frac{\overline{BC}}{5}

As \dpi{120} \overline{AB} + \overline{BC} 21, then, \dpi{120} \overline{AB} 21 - \overline{BC}. Substituting in the above expression, we have:

\dpi{120} \frac{21-\overline{BC}}{2} \frac{\overline{BC}}{5}

Applying the fundamental property of proportions:

\dpi{120} \Rightarrow 2\overline{BC} 5(21- \overline{BC})
\dpi{120} \Rightarrow 2\overline{BC} 105- 5\overline{BC}
\dpi{120} \Rightarrow 7\overline{BC} 105
\dpi{120} \Rightarrow \overline{BC} \frac{105}{7}
\dpi{120} \Rightarrow \overline{BC} 15

Soon \dpi{120} \overline{AB} 21 - \overline{BC} 21 - 15 6.

Resolution of question 4

Making a representative drawing, we can see that \dpi{120} \overline{AB} + \overline{BC} + \overline{AC} 52.

similar triangles

Since the sides of the triangles are proportional, we have:

\dpi{120} \frac{\overline{AB}}{2} \frac{\overline{BC}}{6} \frac{\overline{AC}}{5} r

Being \dpi{120} r the ratio of proportionality.

Furthermore, if the sides are proportional, their sum, that is, the perimeters, are also:

\dpi{120} \frac{\overline{AB} + \overline{BC} +\overline{AC} }{2 + 6 + 5} r
\dpi{120} \Rightarrow \frac{52 }{13} r
\dpi{120} \Rightarrow r 4

From the ratio of proportionality and the known sides, we obtain the measures of the sides of the other triangle:

\dpi{120} \overline{AB} r\cdot \overline{A'B'} 4\cdot 2 8
\dpi{120} \overline{BC} r\cdot \overline{B'C'} 4\cdot 6 24
\dpi{120} \overline{AC} r\cdot \overline{A'C'} 4\cdot 5 20

To download this list of exercises on proportional segments in PDF, click here!

You may also be interested:

  • similarity of triangles
  • Thales Theorem
  • List of exercises on similarity of triangles
  • List of exercises on ratio and proportion
  • List of exercises on Thales' theorem

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