Circle Circuit Worksheet
Circle Circuit
ADVANCED GEOMETRY UNIT 08
Engineer:
Date:
Mission Briefing: Solve each multi-step problem using your knowledge of tangent-radius theorems, chord properties, and arc length formulas. All diagrams are NOT to scale. Show all algebraic steps for full credit. Use \( \pi \approx 3.14159 \) where necessary or leave answers in terms of \( \pi \) as specified.
1
The Tangent Pivot
Line \( \overline{PA} \) is tangent to circle \( O \) at point \( A \). Point \( P \) is located 25 units from the center \( O \). Chord \( \overline{AB} \) is drawn such that it is perpendicular to radius \( \overline{OB} \). If the radius of the circle is 15 units:
- Find the length of tangent \( \overline{PA} \).
- Find the length of chord \( \overline{AB} \).
- Determine the distance from center \( O \) to chord \( \overline{AB} \).
O P A B
Show your calculations here...
C X Y r = 12 cm
2
The Sector Splice
In circle \( C \), chord \( \overline{XY} \) has a length of 12 cm. The radius of the circle is also 12 cm.
- Identify the type of triangle formed by \( \triangle CXY \).
- Calculate the measure of central angle \( \angle XCY \).
- Find the exact length of major arc \( \widehat{XBY} \) (where \( B \) is a point on the larger part of the circle).
Show your calculations here...
3
The Transmission Gears
Two non-overlapping circles with radii 9 and 4 units are positioned such that the distance between their centers \( O_1 \) and \( O_2 \) is 13 units. A common external tangent segment \( \overline{ST} \) touches the circles at points \( S \) and \( T \).
Calculate:
- The length of the tangent segment \( \overline{ST} \).
- If a chord is drawn in the larger circle parallel to \( \overline{ST} \) at a distance of 3 units from center \( O_1 \), what is the length of this chord?
O₁ O₂ S T
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4
The Perimeter Lock
A circle is inscribed in a right trapezoid \( ABCD \). Points \( E, F, G, \) and \( H \) are the points of tangency on sides \( \overline{AB}, \overline{BC}, \overline{CD}, \) and \( \overline{DA} \) respectively. If the radius of the circle is 6 and the length of segment \( \overline{BF} \) is 4, and \( \overline{GC} \) is 9:
- Find the lengths of all sides of the trapezoid.
- Calculate the area of the trapezoid.
- Find the length of the arc \( \widehat{EFG} \).
Show your detailed proof and calculations here...
Circle Circuit Answer Key
Circle Circuit
Solution Key & Teacher Guide
Unit 08: Circles
1
The Tangent Pivot
A) Tangent \( PA \)
In right triangle \( \triangle OAP \), \( OA^2 + PA^2 = OP^2 \).
\( 15^2 + PA^2 = 25^2 \)
\( 225 + PA^2 = 625 \rightarrow PA = \sqrt{400} = \mathbf{20} \text{ units} \).
B) Distance from O to Chord \( AB \)
Chord \( AB \) is perpendicular to radius \( OB \). Points \( A, B \) are on the circle. The distance from \( O \) to \( AB \) is the projection of radius \( OA \) onto \( OB \). This requires trig or coordinate geometry given the current constraints.
Assume \( AB \) is a chord and its distance from center is needed via the tangent relationship.
Mathematical Modeling
20 15
2
The Sector Splice
1. Triangle Type: Since \( CX = CY = 12 \) (radii) and \( XY = 12 \) (chord), \( \triangle CXY \) is equilateral.
2. Central Angle: In an equilateral triangle, all angles are 60°. Thus, \( \angle XCY = \mathbf{60^\circ} \).
3. Major Arc Length: The measure of the major arc is \( 360^\circ - 60^\circ = 300^\circ \).
Length \( = \frac{300}{360} \times 2\pi(12) = \frac{5}{6} \times 24\pi = \mathbf{20\pi} \text{ cm} \approx 62.83 \text{ cm} \).
3
The Transmission Gears
1. Common Tangent: Use the formula \( ST = \sqrt{d^2 - (r_1 - r_2)^2} \).
\( ST = \sqrt{13^2 - (9 - 4)^2} = \sqrt{169 - 25} = \sqrt{144} = \mathbf{12} \text{ units} \).
2. Chord Length: In the circle with radius 9, the chord is 3 units from the center.
Using the perpendicular radius theorem: \( (\frac{1}{2}\text{chord})^2 + 3^2 = 9^2 \).
\( (\frac{1}{2}\text{chord})^2 = 81 - 9 = 72 \).
\( \frac{1}{2}\text{chord} = \sqrt{72} = 6\sqrt{2} \).
Total chord length \( = \mathbf{12\sqrt{2}} \text{ units} \approx 16.97 \text{ units} \).
4
The Perimeter Lock
Step 1: Identify tangent segments.
\( AE = AH = 6 \) (radius)
\( EB = BF = 4 \)
\( FC = CG = 9 \)
\( GD = DH = 6 \) (radius)
Step 2: Calculate side lengths.
\( AB = 6 + 4 = \mathbf{10} \)
\( BC = 4 + 9 = \mathbf{13} \)
\( CD = 9 + 6 = \mathbf{15} \)
\( DA = 6 + 6 = \mathbf{12} \)
Step 3: Area.
Area \( = \frac{1}{2}(AB + CD)h = \frac{1}{2}(10 + 15)12 = \mathbf{150} \text{ units}^2 \).
