Question

1. solve for x: (a) if x = 75° outline the nose, eyes, eyebrows, scar, and mouth in red. (b) if x = 65° outline the nose, eyes, eyebrows, scar, and mouth in black. 20. solve for x: (a) if x = 65° color the shirt collar red. (b) if x = 95° color the shirt collar green. 22. solve for x: (a) if x = 27√7 color the shirt area with the buttons green. (b) if x = 3√21 color the shirt area with the buttons yellow. 23. solve for x: (a) if x = 4 color the rest of the shirt orange. (b) if x = 8 color the rest of the shirt black.

Explanation:

Problem 22: Solve for \( x \)

We can use the Pythagorean theorem in the right triangle formed. The radius of the circle is \( x \), the length of the tangent segment is \( 10 \), and the length from the external point to the center is \( 17 \). For a tangent to a circle, the radius is perpendicular to the tangent, so we have a right triangle with hypotenuse \( 17 \), one leg \( x \) (radius), and the other leg \( 10 \) (tangent segment).

Step 1: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \( a \), \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). Here, let \( a = x \), \( b=10 \) and \( c = 17 \). So we have the equation:

$$ x^{2}+10^{2}=17^{2} $$

Step 2: Solve for \( x^{2} \)

First, calculate \( 10^{2}=100 \) and \( 17^{2} = 289 \). Then:

$$ x^{2}+ 100=289 $$

Subtract \( 100 \) from both sides:

$$ x^{2}=289 - 100=189 $$

Step 3: Simplify \( x \)

We can factor \( 189=9\times21 \), so:

$$ x=\sqrt{189}=\sqrt{9\times21} = 3\sqrt{21} $$

Problem 23: Solve for \( x \)

We can use the property of tangents and secants. If we have a secant segment and a tangent segment from an external point to a circle, the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part. Let the radius of the circle be \( x \), so the diameter is \( 2x \), the length of the tangent segment is \( 6 \), the external part of the secant is \( 2 \), and the entire secant segment is \( 2 + 2x\) (since the internal part of the secant is the diameter \( 2x\)).

Step 1: Apply the tangent - secant rule

The tangent - secant rule states that if a tangent of length \( t \) and a secant with external part \( e \) and internal part \( i \) are drawn from an external point to a circle, then \( t^{2}=e\times(e + i) \). Here, \( t = 6 \), \( e=2 \) and \( i = 2x \). So we have:

$$ 6^{2}=2\times(2 + 2x) $$

Step 2: Simplify the equation

First, calculate \( 6^{2}=36 \). Then:

$$ 36=2\times(2 + 2x) $$

Divide both sides by \( 2 \):

$$ 18=2 + 2x $$

Step 3: Solve for \( x \)

Subtract \( 2 \) from both sides:

$$ 2x=18 - 2=16 $$

Divide both sides by \( 2 \):

$$ x = 8 $$

Final Answers

• For problem 22: \( x = 3\sqrt{21}\)
• For problem 23: \( x=8\)

Answer:

Problem 22: Solve for \( x \)

We can use the Pythagorean theorem in the right triangle formed. The radius of the circle is \( x \), the length of the tangent segment is \( 10 \), and the length from the external point to the center is \( 17 \). For a tangent to a circle, the radius is perpendicular to the tangent, so we have a right triangle with hypotenuse \( 17 \), one leg \( x \) (radius), and the other leg \( 10 \) (tangent segment).

Step 1: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \( a \), \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). Here, let \( a = x \), \( b=10 \) and \( c = 17 \). So we have the equation:

$$ x^{2}+10^{2}=17^{2} $$

Step 2: Solve for \( x^{2} \)

First, calculate \( 10^{2}=100 \) and \( 17^{2} = 289 \). Then:

$$ x^{2}+ 100=289 $$

Subtract \( 100 \) from both sides:

$$ x^{2}=289 - 100=189 $$

Step 3: Simplify \( x \)

We can factor \( 189=9\times21 \), so:

$$ x=\sqrt{189}=\sqrt{9\times21} = 3\sqrt{21} $$

Problem 23: Solve for \( x \)

We can use the property of tangents and secants. If we have a secant segment and a tangent segment from an external point to a circle, the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part. Let the radius of the circle be \( x \), so the diameter is \( 2x \), the length of the tangent segment is \( 6 \), the external part of the secant is \( 2 \), and the entire secant segment is \( 2 + 2x\) (since the internal part of the secant is the diameter \( 2x\)).

Step 1: Apply the tangent - secant rule

The tangent - secant rule states that if a tangent of length \( t \) and a secant with external part \( e \) and internal part \( i \) are drawn from an external point to a circle, then \( t^{2}=e\times(e + i) \). Here, \( t = 6 \), \( e=2 \) and \( i = 2x \). So we have:

$$ 6^{2}=2\times(2 + 2x) $$

Step 2: Simplify the equation

First, calculate \( 6^{2}=36 \). Then:

$$ 36=2\times(2 + 2x) $$

Divide both sides by \( 2 \):

$$ 18=2 + 2x $$

Step 3: Solve for \( x \)

Subtract \( 2 \) from both sides:

$$ 2x=18 - 2=16 $$

Divide both sides by \( 2 \):

$$ x = 8 $$

Final Answers

• For problem 22: \( x = 3\sqrt{21}\)
• For problem 23: \( x=8\)