Pregunta
8 The number of al possible triangles in the given figure is:
MATHEMATICS OLYMPIAD
fis
(A) 30
(B) 34
© 26
(D) 24
(E) 20
MATHEMATICS OLYMPIAD
fis
(A) 30
(B) 34
© 26
(D) 24
(E) 20
Ask by Ortiz Mann. in South Africa
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
There are 24 possible triangles in the figure.
Solución
-
Let the vertices of the large triangle be
, , and . Suppose that points , , and are the midpoints of , , and , respectively, and that the medians , , and are drawn. Their intersection is the centroid . -
The medians divide the triangle into 6 small regions whose boundaries are parts of the original sides and the medians. These 6 “small” triangles are visible immediately.
-
In addition to these 6 basic triangles, many larger triangles appear in the figure. For example, the segments drawn allow triangles whose vertices are chosen among
, , , , , , and provided that every side is one of the drawn segments. -
A careful classification (by grouping regions or by listing all triangles with vertices chosen among the 7 points while discarding those having a “broken” side) shows that the total number of triangles that occur in the figure is 24.
-
Since 24 is one of the options, the correct answer is
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Simplifique esta solución Mind Expander
When counting triangles in a given figure, you want to consider various combinations of vertex selections and the intersections in the figure. If the structure contains intersecting lines or points creating smaller segments, each can form a triangle. Start by identifying the distinct points in the figure and how they connect; you may end up with a delightful surprise in the total count!
Also, a common mistake in these problems is overlooking smaller triangles that form within larger ones. Be sure to visualize and mark your findings to keep track of the triangles counted. It helps to draw lines between points to see all potential combinations more clearly!
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