Question

AC and DB are chords that intersect at point H.
What is the length of line segment DB?
4 units
8 units
16 units
20 units

Answer 1

Answer:

16 units

Step-by-step explanation:

see the attached figure to better understand the problem

Applying  the Intersecting Chords Theorem

we have that

$AH*HC=DH*HB$

substitute the given values

$(20-x)(x)=(x+4)(12-x)$

$20x-x^2=12x-x^2+48-4x$

Simplify

$20x=8x+48$

$20x-8x=48$

$12x=48$

$x=4$

Find out the length of segment DB

$DB=DH+HB$

$DB=(x+4)+(12-x)=16\ units$

Note In this problem it was not necessary to determine the value of x to calculate the DB segment.

Anyway, the calculation was done for didactic purposes.

Answer 2

The length of line segment DB is 16 units

Explanation:

The Intersecting Chords Theorem or chord theorem states that when two chords are intersecting within a circle, the product of their respective segments is equal. This also a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords in a circle. It states that the products of the lengths of the line segments on each chord are equal.

AC and DB are chords that intersect at point H as shown in the figure below.

The intersection definition is, if the intersection happened of two sets A and B it is denoted by A ∩ B. It is the set  that contains all elements of A that also belong to B, and nothing else. An intersection is a single point where two lines meet or cross each other.

$AH*HC=DH*HB\\ (20-x)*(x)=(x+4)*(12-x)\\ 20x-x^{2} =12x-x^{2} +48-4x\\ 20x-12x+4x=48\\ 12x=48\\ x=4$

Then we should find the length of the segment DB . The line segment is a part of a line with two distinct end points.

$DB=(x+4)+(12-x)\\ DB=(4+4)+(12-4)\\ DB=8+8\\ DB=16\ units$

Therefore  the length of line segment DB is equal to  $16$ units

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