Question

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3. Is AB tangent to circle C? Explain.

Answer 1

Question 3, part (a)

The tangent line is perpendicular to the radius at the point of tangency.

If AB was tangent to circle C, then radius AC would be perpendicular to segment AB. Meaning that angle BAC should be 90 degrees, and it should lead to triangle ABC being a right triangle.

Use the pythagorean theorem converse to see if we have a right triangle or not.

a^2 + b^2 = c^2

5^2 + 11^2 = 13^2

25 + 121 = 169

146 = 169

The last equation is false so the first equation is false when a = 5, b = 11, c = 13. Triangle ABC is not a right triangle. We've proven angle BAC isn't 90 degrees.

Answer: Not tangent

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Question 3, part (b)

We'll use the same template of steps as done back in part (a).

This time we have:

• a = 6
• b = 8
• c = 10

The order of 'a' and b doesn't matter. The c is always the longest side.

a^2 + b^2 = c^2

6^2 + 8^2 = 10^2

36 + 64 = 100

100 = 100

The last result is true so the first equation is true for the a,b,c values mentioned. Notice the equation would also be true if a = 8, b = 6, c = 10.

Therefore, triangle ABC is a right triangle where the 90 degree angle is at point A.

Answer: AB is tangent to the circle.

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Summary

• 3(a) Not tangent
• 3(b) Tangent

Answer 2

(a) No

(b) Yes

Explanation:

The tangent of a circle is always perpendicular to the radius.

Since AC is the radius of circle C, if AB is a tangent to circle C, then m∠BAC = 90°.

As we have been given the measures of all three sides of both triangles, we can use Pythagoras Theorem to determine if the measure of angle BAC in both triangles is a right angle.

Pythagoras Theorem explains the relationship between the three sides of a right triangle. The square of the hypotenuse (longest side) is equal to the sum of the squares of the legs of a right triangle:

`\boxed{a^2+b^2=c^2}`

where:

• a and b are the legs of the right triangle.
• c is the hypotenuse (longest side) of the right triangle.

Part (a)

The legs of this triangle are AC and AB. The hypotenuse is BC.

Substitute the measures of the sides into the Pythagoras Theorem formula:

`\implies AC^2+AB^2=BC^2`

`\implies 5^2+11^2=13^2`

`\implies25+121=169`

`\implies146=169`

As 146 does not equal 169, this proves that the triangle is not a right triangle. Therefore AB is not tangent to circle C.

Part (b)

The legs of this triangle are AC and AB. The hypotenuse is BC.

Substitute the measures of the sides into the Pythagoras Theorem formula:

`\implies AC^2+AB^2=BC^2`

`\implies 6^2+8^2=10^2`

`\implies 36+64=100`

`\implies100=100`

As both sides of the equation are the same, this proves that the triangle is a right triangle and m∠BAC = 90°. Therefore AB is tangent to circle C.