The chords cd and ab intersect at the point O, CO = 4 cm, OD = 3cm, Paoc = 9cm, and the segment AO is three times smaller than the segment OB. Calculate chord length DB
Given: chords, where О ∈ AB, О ∈ CD, СО = 4 cm; OD = 3 cm, PΔAOC = 9 cm and ОВ = 3 * AO. It is necessary to find the length of the chord DВ.
We will use the following property of intersecting chords. The products of the lengths of the segments into which each of the chords is divided are equal. This fact for our assignment can be formalized as follows: AO * BO = CO * DO. We have AO * (3 * AO) = (4 cm) * (3 cm) or AO ^ 2 = 4 cm2, whence AO = 2 cm.
Since PΔAOC = AO + CO + AC = 9 cm, then AC = 9 cm – AO – CO = 9 cm – 2 cm – 4 cm = 3 cm.
It is easy to prove that the triangles AOC and BOD are similar. Indeed, the angles AOC and BOD are equal as vertical angles. Angles B and C are based on the same arc AD, therefore, they are equal. Likewise, angles A and D are equal, as being supported by one arc BC. According to the I sign of similarity of triangles, triangles AOC and BOD are similar.
According to the properties of such triangles: AC: DB = AO: OD. We have (3 cm): DB = (2 cm): (3 cm). Let us consider the last equality as a proportion. Then (2 cm) * DB = (3 cm) * (3 cm), whence DB = (9: 2) cm = 4.5 cm.
Answer: The chord length DB is 4.5 cm.
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