In trapezoid ABCD it is known that AD = 18cm, BC = 14cm, AC = 24cm. Find the line segments into which the AC diagonal

| June 25, 2021 | education

In trapezoid ABCD it is known that AD = 18cm, BC = 14cm, AC = 24cm. Find the line segments into which the AC diagonal is divided by the point of intersection of the diagonals.

1. Diagonals AC and BD intersect at point O.

2. ∠САD = ∠АСВ as internal criss-crossing, formed by the parallel bases of the trapezoid and the diagonal AC.

3. ∠СВD = ∠ADВ as internal criss-crossing formed by the parallel bases of the trapezoid and the diagonal ВD.

4. Therefore, triangles BOC and AOD are similar in two equal angles.

5. According to the properties of vertical angles ∠ВОС = ∠АОD.

6. Similar sides (opposite equal angles) of similar triangles are proportional. Therefore, BC / AD = CO / AO.

7. Let’s designate the length of the segment with the index “a”. The length of the segment AC (24 – a).

8. BC / AD = a / (24 – a);

14/18 = a / (24 – a);

14 x 24 – 14a = 18a;

32a = 336;

a = 10.5 cm is the length of the CO segment.

The length of the segment AO = 24 – a = 24 – 10.5 = 13.5 cm.

Answer: the length of the CO segment is 10.5 cm, the length of the AO segment is 13.5 cm.



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