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Given that ๐ธ๐ถ equals four, ๐ธ๐ท equals 15, and ๐ธ๐ต equals six, find the length of the line segment ๐ธ๐ด.
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Letโs look at the diagram more closely.
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We can see that it consists of a circle, in which there are two intersecting chords.
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Theyโre the lines ๐ด๐ต and ๐ถ๐ท.
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Weโve also been given various lengths that we can add to our diagram.
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The length of the line segment ๐ธ๐ถ is four.
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The length of the line segment ๐ธ๐ท is 15.
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And the length of the line segment ๐ธ๐ต is six.
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Weโre asked to work out the length of the line segment ๐ธ๐ด.
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So we need to recall the relationship that exists between the lengths of the line segments of intersecting chords.
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We remember that โif two chords intersect in a circle, then the products of the lengths of the chord segments are equal.โ
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Our two chords intersect inside the circle at the point ๐ธ.
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So we have that the product of the lengths of the chord segments of the orange chord, thatโs ๐ธ๐ด multiplied by ๐ธ๐ต, is equal to the product of the length of the chord segment of the pink cord.
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Thatโs ๐ธ๐ถ multiplied by ๐ธ๐ท.
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We know the lengths of the chord segments ๐ธ๐ต, ๐ธ๐ถ, and ๐ธ๐ท.
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So we can substitute their values into this equation given that the length of the chord segment ๐ธ๐ด multiplied by six is equal to four multiplied by 15.
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Dividing both sides of this equation through by six gives a calculation that we can use to find the length of the chord segment ๐ธ๐ด.
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Itโs equal to four multiplied by 15 over six.
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Four multiplied by 15 is equal to 60.
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And 60 divided by six is equal to 10.
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Using then the relationship between the lengths of chord segments for chords which intersect inside a circle, we found that the length of the chord segment ๐ธ๐ด is 10.
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Thereโre no units for this as there were no units for the original measurements we were given for the other three chord segments.