Dan BRÂNZEI : Inside is Geometry! Those who love it and those who don't are invited in!

 

Evaluarea nationala

          At the National Assessment for VIII-th grade graduates, the students received the following subjects: see the section "Subiecte si bareme" (Subjects and scales).

          Regarding the Mathematics test, the subject is in the

 "ENVIII_Matematica_2024_var_07_LRO.pdf" 

file, and the correction scale in the 

"ENVIII_Matematica_2024_bar_07_LRO.pdf"

file. There was also, like every year, a reserve option : see "Matematică rezervă - 27 iunie 2024" section; see files "ENVIII_Matematica_2024_var_02_LRO.pdf" and "ENVIII_Matematica_2024_bar_02_LRO.pdf".

          Two problems were considered more difficult. Problem III.4 and problem III.5.

In translation: 

                         "The attached figure shows the isosceles triangle $ABC$ with $AB=AC$.

                       The height from the top $A$ intersects the side $BC$ at point $D$ and $AD=BC$. 

                       The height from vertex $B$ intersects side $AC$ at point $E$. Heights $AD$

                        and $BE$ intersect at point $H$.

                          a) Show that angles $DAC$ and $EBC$ have the same measure.

                          b) Prove that $AH=3 \cdot HD$."

In translation:

                         "The attached figure shows the circle with center $O$, in which $CD$ is

                          the diameter. Point $B$ belongs to the circle so that lines $BO$ and $CD$

                          are perpendicular. Point $M$ belongs to the small arc $BC$, lines $DM$

                          and $BO$ intersect at point $N$, $DN=2 \cdot MN$ and $MN=4\;cm$.

                               a)  Show that the measure of the angle $CMD$ is $90^{\circ}$.

                               b)  Calculate the area of ​​the triangle $DON$."

               The full Subject and Scale are in the Images below.

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                    Solution and comments CiP

                The solution to exercise III.4.a) is standard, noting that the angle $\measuredangle DAC$ is, in the right triangle $ACD$, the complement of the angle $\measuredangle C$, and the angle $\measuredangle EBC$ is, in the right triangle $BCE$, the complement of the same angle $\measuredangle C$.

          In exercise III.4.b) the key to the solution is the consideration of similar triangles $ACD$ and $BDH$. The order of the vertices in the relation of similarity is very important: respectively equal angles are opposed to respectively proportional sides. In more detail,

          first, $BD=DC=\frac{BC}{2}=\frac{AD}{2}$;

          secondly $\measuredangle BDH =90^{\circ}=\measuredangle ADC$ and 

$$\measuredangle HBD \underset{same\;angle}{=}\measuredangle EBC \underset{a)}{=}\measuredangle CAD.$$

Then from the last row above we have based on the "AA" criterion that

$$\Delta \;BDH \sim \Delta \;ADC$$

hence

$$\frac{BD}{AD}=\frac{DH}{DC}=\frac{BH}{AC}. \tag{$\sigma$}$$

But $\frac{BD}{AD}=\frac{\frac{BC}{2}}{BC}=\frac{1}{2}$, so from ($\sigma$) $\frac{HD}{DC}=\frac{1}{2}$. From here $HD=\frac{DC}{2}=\frac{BC}{4}=\frac{AD}{4}$.

          We get $AH=AD-HD=AD-\frac{AD}{4}=3 \cdot \frac{AD}{4}=3 \cdot HD.$

$\square$

               The key to exercise III.5.b) consists in determining the radius of the circle. For this, all the solutions I have come across are based on writing segments ratios. The most common is $\frac{OD}{ND}=\frac{MD}{CD}$. In the official scale, it results from writing the cosine of angle $\measuredangle C$ in two different triangles. Or one could use, less common with our students, the similarity of the triangles $DNO$ and $DCM$.

          But a simple idea can solve the problem without resorting to the most elementary knowledge of geometry. Join points C and M.

          $BO$ is the median of the segment $[CD]$, so the point $N$ on it verifies $NC=ND$. Hence 

$$NC=8.$$

          The right triangle $CMN$ has the leg $MN$ half of the hypotenuse $CN$ therefore it is a "30-60-90" triangle. (I have marked this on the figure with blue lines.) We still have $\measuredangle CND=180^{\circ}-60^{\circ}=120^{\circ}$. But in the isosceles triangle $CDN$, the median $NO$ from the vertex $N$ is also a bisector, so $\measuredangle CNO =\measuredangle DON =120^{\circ}\;:\;2=60^{\circ}$ (marked with red). It follows easily that the triangles $CNO$ and $DNO$ are of the same type. So $NO=4$ and we now have enough data to finish the problem.

$\blacksquare$

Juliusz Brzeziński Galois teori genom övningar // Galois Theory Through Exercises by Juliusz Brzeziński

                A source where you can download this book is here.

 

                I read an earlier version ("2015"), written for students. 

                 An interesting problem caught my attention: on page 102, problem 16.79 (in the "2015" version it is page 87, problem 16.40). But without indications of a solution. I have made posts with this type of problem here, here and here.