Construct two intersecting chords inside a circle and measure the parts of the segments formed by the intersecting chords. Then, construct two secants through a point outside the circle and measure the segments formed.
See solution.
Practice makes perfect
We will examine the relationship among the segments formed by two intersecting chords. Then, we will examine the relationship among the segments of two secants that intersect outside a circle.
Segments Formed by Two Intersecting Chords
When two chords intersect inside a circle, they form four smaller segments with a common endpoint. Let's examine the graph that shows the length of each segment formed by the intersecting chords.
We have three cases to consider. We will make a table to calculate the product of the lengths of the parts of each chord.
BF * CF
DF * EF
Case I
2.46 * 3.18 ≈ 7.82
3.80 * 2.06 ≈ 7.83
Case II
3.33 * 2.31 ≈ 7.69
3.88 * 1.98 ≈ 7.68
Case III
4.33 * 1.31 ≈ 5.67
2.43 * 2.33 ≈ 5.66
For each case the products are approximately equal. They are not exactly the same because, instead of using exact values, we used approximated values. Therefore, we can conclude that when two chords intersect inside of a circle, the product of the lengths of the segments formed by each chord are equal.
BF * CF = DF * EF
Secants Intersecting Outside a Circle
Next, we will check whether there exists a relationship among the segments of two secants that intersect outside a circle. Let's investigate the diagram below.
Let's make a table to calculate the product of the segments for each case.
BE * BC
BF * BD
Case I
2.19* 7.34 ≈ 16.07
2.07* 7.77 ≈ 16.08
Case II
4.90* 8.99 ≈ 44.05
4.30* 10.23 ≈ 43.99
Case III
3.58* 9.29 ≈ 33.26
3.83* 8.70 ≈ 33.32
The products are approximately equal. Again, they are not exactly the same because we used approximations. Therefore, for a secant line, the product of the lengths of the segments with endpoints at the point outside the circle and the intersection with the circle are equal.
BE * BC = BF * BD
