11-Tangents to Circles - Kuta Software - Free Printable

This image is a worksheet on "Tangents to Circles" from Kuta Software. The task involves two main sections:

1. Determine if line AB is tangent to the circle (Problems 1-4).
2. Find the segment length indicated (Problems 5-8).

I will solve each problem step-by-step.

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Section 1: Determine if line AB is tangent to the circle

A line is tangent to a circle if it intersects the circle at exactly one point and is perpendicular to the radius at that point. We can use the Pythagorean theorem to check this. If the triangle formed by the center of the circle, the point of tangency, and the external point is a right triangle, then the line is tangent.

#### Problem 1:
- Triangle sides: 8, 12, 16
- Check if $8^2 + 12^2 = 16^2$
- $64 + 144 = 208$, but $16^2 = 256$
- Since $208 \neq 256$, it is not a right triangle.
- Therefore, line AB is not tangent to the circle.

#### Problem 2:
- Triangle sides: 9.6, 11, 13
- Check if $9.6^2 + 11^2 = 13^2$
- $92.16 + 121 = 213.16$, but $13^2 = 169$
- Since $213.16 \neq 169$, it is not a right triangle.
- Therefore, line AB is not tangent to the circle.

#### Problem 3:
- Triangle sides: 12, 16, 20
- Check if $12^2 + 16^2 = 20^2$
- $144 + 256 = 400$, and $20^2 = 400$
- Since $400 = 400$, it is a right triangle.
- Therefore, line AB is tangent to the circle.

#### Problem 4:
- Triangle sides: 11.4, 15.2, 19
- Check if $11.4^2 + 15.2^2 = 19^2$
- $129.96 + 231.04 = 361$, and $19^2 = 361$
- Since $361 = 361$, it is a right triangle.
- Therefore, line AB is tangent to the circle.

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Section 2: Find the segment length indicated

We assume lines that appear to be tangent are tangent. We can use the Pythagorean theorem or properties of tangents to find the unknown lengths.

#### Problem 5:
- Right triangle with legs 4 and 8.5, hypotenuse unknown.
- Let the unknown side be $x$.
- $4^2 + 8.5^2 = x^2$
- $16 + 72.25 = x^2$
- $88.25 = x^2$
- $x = \sqrt{88.25} \approx 9.4$

#### Problem 6:
- Right triangle with legs 3.5 and 1, hypotenuse unknown.
- Let the unknown side be $x$.
- $3.5^2 + 1^2 = x^2$
- $12.25 + 1 = x^2$
- $13.25 = x^2$
- $x = \sqrt{13.25} \approx 3.64$

#### Problem 7:
- Right triangle with legs 12 and 16, hypotenuse unknown.
- Let the unknown side be $x$.
- $12^2 + 16^2 = x^2$
- $144 + 256 = x^2$
- $400 = x^2$
- $x = \sqrt{400} = 20$

#### Problem 8:
- Right triangle with legs 6 and 6.4, hypotenuse unknown.
- Let the unknown side be $x$.
- $6^2 + 6.4^2 = x^2$
- $36 + 40.96 = x^2$
- $76.96 = x^2$
- $x = \sqrt{76.96} \approx 8.77$

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Final Answers:

Section 1:
1. Not tangent
2. Not tangent
3. Tangent
4. Tangent

Section 2:
5. $\approx 9.4$
6. $\approx 3.64$
7. $20$
8. $\approx 8.77$

These solutions are based on the geometric properties of tangents and the Pythagorean theorem.