PDF) Arcs and Chords | Mark Ortego - Academia.edu - Free Printable

To find the length of the indicated segment $x$ in each problem, we use the properties of right triangles formed by chords and radii/diameters in a circle.

Key Concept:
When a radius or diameter is perpendicular to a chord, it bisects the chord (cuts it into two equal halves). This creates a right-angled triangle where:
1. The hypotenuse is the radius ($r$) or part of a diameter.
2. One leg is half the length of the chord.
3. The other leg is the distance from the center to the chord (or part of the radius/diameter).

We use the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

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Problem 1

* Given: A radius of $7.1$ and half of a chord of length $6$. We need to find $x$, which is the other leg of the right triangle.
* Setup: The triangle has legs $x$ and $6$, and hypotenuse $7.1$.
* Calculation:
$$x^2 + 6^2 = 7.1^2$$
$$x^2 + 36 = 50.41$$
$$x^2 = 50.41 - 36$$
$$x^2 = 14.41$$
$$x = \sqrt{14.41}$$
$$x \approx 3.8$$

Problem 2

* Given: A radius of $3.2$ and half of a chord of length $3$. We need to find $x$, the distance from the center to the chord.
* Setup: The triangle has legs $x$ and $3$, and hypotenuse $3.2$.
* Calculation:
$$x^2 + 3^2 = 3.2^2$$
$$x^2 + 9 = 10.24$$
$$x^2 = 10.24 - 9$$
$$x^2 = 1.24$$
$$x = \sqrt{1.24}$$
$$x \approx 1.1$$

Problem 3

* Given: A radius of $31.8$ and a segment from the center to the chord of length $7$. We need to find $x$, which is half the chord.
* Setup: The triangle has legs $7$ and $x$, and hypotenuse $31.8$.
* Calculation:
$$7^2 + x^2 = 31.8^2$$
$$49 + x^2 = 1011.24$$
$$x^2 = 1011.24 - 49$$
$$x^2 = 962.24$$
$$x = \sqrt{962.24}$$
$$x \approx 31.0$$

Problem 4

* Given: Two chords intersecting at the center? No, looking closely, one line is a diameter because it passes through the center point (indicated by the dot). The segments on the diameter are $9.7$ and $9.7$, so the radius is $9.7$. The other chord is perpendicular to the diameter. The segment labeled $5.1$ is part of the diameter from the center to the chord. Wait, let's re-examine.
* The line passing through the center has segments $9.7$ and $9.7$. This confirms the radius $r = 9.7$.
* The perpendicular chord is bisected. Let half of it be $h$.
* The segment labeled $5.1$ is the distance from the center to the chord.
* We need to find $x$, which is half of the perpendicular chord.
* Setup: Right triangle with legs $5.1$ and $x$, and hypotenuse $9.7$ (the radius).
* Calculation:
$$5.1^2 + x^2 = 9.7^2$$
$$26.01 + x^2 = 94.09$$
$$x^2 = 94.09 - 26.01$$
$$x^2 = 68.08$$
$$x = \sqrt{68.08}$$
$$x \approx 8.3$$

Problem 5

* Given: A diameter is shown with total length $25.2 + 4.5$? No, let's look at the labels.
* There is a radius labeled $25.2$? No, the segment from the center to the edge is $25.2$. So radius $r = 25.2$.
* There is a chord perpendicular to a radius.
* The segment from the center to the chord is labeled $4.5$? No, looking at the diagram, the segment labeled $4.5$ is part of the radius perpendicular to the chord. The segment labeled $x$ is half of the chord.
* Wait, let's look closer. The line going through the center is a diameter. The segment from the center to the circle edge is the radius. One part is labeled $25.2$. Is that the whole radius? Or is $25.2$ the segment from the center to the other side?
* Let's re-read the diagram. The dot is the center. The segment from the center to the right edge is $25.2$. So, Radius $r = 25.2$.
* The perpendicular segment from the center to the chord is labeled $4.5$? No, the label $4.5$ is on the segment from the chord to the circle edge along the radius.
* Let's assume the standard configuration: The radius is the sum of the inner part and outer part. If the outer part (from chord to circle) is $4.5$, and the total radius is $25.2$, then the distance from the center to the chord is $25.2 - 4.5 = 20.7$.
* We need to find $x$, which is half the chord.
* Hypotenuse = Radius = $25.2$.
* Leg 1 = Distance from center to chord = $25.2 - 4.5 = 20.7$.
* Leg 2 = $x$.
* Calculation:
$$20.7^2 + x^2 = 25.2^2$$
$$428.49 + x^2 = 635.04$$
$$x^2 = 635.04 - 428.49$$
$$x^2 = 206.55$$
$$x = \sqrt{206.55}$$
$$x \approx 14.4$$

*Alternative Interpretation:* What if $25.2$ is the diameter? Then radius is $12.6$. If the small segment is $4.5$, the distance to center is $12.6 - 4.5 = 8.1$.
$8.1^2 + x^2 = 12.6^2 \rightarrow 65.61 + x^2 = 158.76 \rightarrow x^2 = 93.15 \rightarrow x \approx 9.7$.
Looking at the visual proportions, the segment labeled $25.2$ starts at the center and goes to the edge. It looks like a radius. The segment $4.5$ is the "cap" from the chord to the edge. So the first interpretation ($r=25.2$) is most likely correct given typical worksheet layouts where long lines from center are radii. However, let's look at Problem 6 for consistency. In Problem 6, $18.8$ is clearly a radius. In Problem 5, $25.2$ is similarly placed. So $r=25.2$ is correct.

Problem 6

* Given: Radius $r = 18.8$. A chord is perpendicular to a radius. The segment from the chord to the circle edge is $4.2$. We need to find $x$, which is half the chord.
* Setup:
* Hypotenuse = Radius = $18.8$.
* Distance from center to chord = Radius - outer segment = $18.8 - 4.2 = 14.6$.
* Leg = $x$.
* Calculation:
$$14.6^2 + x^2 = 18.8^2$$
$$213.16 + x^2 = 353.44$$
$$x^2 = 353.44 - 213.16$$
$$x^2 = 140.28$$
$$x = \sqrt{140.28}$$
$$x \approx 11.8$$

Final Answer:
1) 3.8
2) 1.1
3) 31.0
4) 8.3
5) 14.4
6) 11.8