Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

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Triangle Inequality Theorem : Can you make a triangle ? Yes! 1

Transcript of Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Page 1: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Triangle Inequality Theorem:

Can you make a triangle?

Yes!

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Page 2: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Triangle Inequality Theorem:

Can you make a triangle?

NObecause

4 + 5 < 12

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Page 3: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Triangle Inequality Theorem:

a

cb

C

A

B

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

a + b > ca + c > bb + c > a

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Page 4: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Finding the range of the third side:Example Given a triangle with sides of length 3 and 7, find the range of possible values for the third side.

Solution Let x be the length of the third side of the triangle.

The maximum value:x < 3 + 7 = 10

The minimum value:x > 7 – 3 = 4

So 4 < x < 10 (x is between 4 and 10.)

x

x

x < 10

x > 4

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Page 5: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Finding the range of the third side:

Given The lengths of two sides of a triangle

Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides.

Since the third side and the smallest side given cannot be larger than the other side, we find the minimum value by subtracting the two sides.

Difference < Third Side < Sum

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Page 6: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Finding the range of the third side:Example Given a triangle with sides of length a and b, find the range of possible values for the third side.

Solution Let x be the length of the third side of the triangle.

The maximum value:x < a + b

The minimum value:x > |a – b|

So |a – b|< x < a + b (x is between |a – b| and a + b.)

x < a + b

x > |a – b|

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Page 7: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

In a Triangle:

The largest angle is opposite the largest side.

mBAC = 36°

mCBA = 80°

mBCA = 64°

m CA = 6.9 cm

m BC = 4.1 cm

m AB = 6.3 cm

A

B

C

The smallest angle is opposite the smallest side.

The smallest side is opposite the smallest angle. The largest side is opposite the largest angle.

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Page 8: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Theorem•If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle.

smaller angle

larger

angle

longer side

shorter side

A

B

C

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Page 9: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Theorem•If one side of a triangle is larger than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

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Page 10: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Corollary #1:

The perpendicular segment from a point to a line is the shortest segment from the point to the line.

This is the shortest segment!

This side is longer because it is opposite the largest angle!

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Page 11: Triangle Inequality Theorem: Can you make a triangle? Yes! 1.

Corollary #2:The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

This is the shortest segment!

This side is longer because it is opposite the largest angle!

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