tr i gonometry

MPM2D: Principles of Mathematics

The Pythagorean Theorem

J. Garvin

Slide 1/10

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Pythagorean Theorem

Consider four congruent triangles with arms a and b andhypotenuses c . They can be arranged in many ways,including the two below.

The white area on the left, c2, is the same as the sum of thewhite areas on the right, a2 + b2.

J. Garvin — The Pythagorean Theorem

Slide 2/10

tr i gonometry

Pythagorean Theorem

Consider four congruent triangles with arms a and b andhypotenuses c . They can be arranged in many ways,including the two below.

The white area on the left, c2, is the same as the sum of thewhite areas on the right, a2 + b2.

J. Garvin — The Pythagorean Theorem

Slide 2/10

tr i gonometry

Pythagorean Theorem

Pythagorean Theorem

If a and b are the arms in right triangle ABC , and c is thehypotenuse, then a2 + b2 = c2.

J. Garvin — The Pythagorean Theorem

Slide 3/10

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Pythagorean Theorem

Example

Determine |AB| in the triangle below.

|AC |2 + |BC |2 = |AB|2

52 + 122 = |AB|2

169 = |AB|2√

169 = |AB|13 = |AB|

J. Garvin — The Pythagorean Theorem

Slide 4/10

tr i gonometry

Pythagorean Theorem

Example

Determine |AB| in the triangle below.

|AC |2 + |BC |2 = |AB|2

52 + 122 = |AB|2

169 = |AB|2√

169 = |AB|13 = |AB|

J. Garvin — The Pythagorean Theorem

Slide 4/10

tr i gonometry

Pythagorean Theorem

Example

Determine |DF | in the triangle below.

|DF |2 + |EF |2 = |DE |2

|DF |2 + 52 = 82

|DF |2 = 64− 25

|DF | =√

39

|DF | ≈ 6.245

J. Garvin — The Pythagorean Theorem

Slide 5/10

tr i gonometry

Pythagorean Theorem

Example

Determine |DF | in the triangle below.

|DF |2 + |EF |2 = |DE |2

|DF |2 + 52 = 82

|DF |2 = 64− 25

|DF | =√

39

|DF | ≈ 6.245

J. Garvin — The Pythagorean Theorem

Slide 5/10

tr i gonometry

Pythagorean Theorem

Example

Determine |GH| in the triangle below.

|GH|2 + |GI |2 = |HI |2

2|GH|2 = 102

2|GH|2 = 100

|GH|2 = 50

|GH| =√

50

|GH| ≈ 7.071

J. Garvin — The Pythagorean Theorem

Slide 6/10

tr i gonometry

Pythagorean Theorem

Example

Determine |GH| in the triangle below.

|GH|2 + |GI |2 = |HI |2

2|GH|2 = 102

2|GH|2 = 100

|GH|2 = 50

|GH| =√

50

|GH| ≈ 7.071

J. Garvin — The Pythagorean Theorem

Slide 6/10

tr i gonometry

Pythagorean Theorem

Example

Determine the area of ∆ABC below.

To determine the area of∆ABC , we first need todetermine its height.

Since the triangle isisosceles, the height willbisect AB at 90◦.

J. Garvin — The Pythagorean Theorem

Slide 7/10

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Pythagorean Theorem

Example

Determine the area of ∆ABC below.

To determine the area of∆ABC , we first need todetermine its height.

Since the triangle isisosceles, the height willbisect AB at 90◦.

J. Garvin — The Pythagorean Theorem

Slide 7/10

tr i gonometry

Pythagorean Theorem

Example

Determine the area of ∆ABC below.

To determine the area of∆ABC , we first need todetermine its height.

Since the triangle isisosceles, the height willbisect AB at 90◦.

J. Garvin — The Pythagorean Theorem

Slide 7/10

tr i gonometry

Pythagorean Theorem

|AD|2 + |CD|2 = |AC |2

42 + |CD|2 = 112

|CD|2 = 105

|CD| =√

105

|CD| ≈ 10.247

Using the formula A = 12bh, the area of ∆ABC is

A = 12 × 8×

√105 ≈ 40.988 square units.

J. Garvin — The Pythagorean Theorem

Slide 8/10

tr i gonometry

Pythagorean Theorem

|AD|2 + |CD|2 = |AC |2

42 + |CD|2 = 112

|CD|2 = 105

|CD| =√

105

|CD| ≈ 10.247

Using the formula A = 12bh, the area of ∆ABC is

A = 12 × 8×

√105 ≈ 40.988 square units.

J. Garvin — The Pythagorean Theorem

Slide 8/10

tr i gonometry

Pythagorean Theorem

|AD|2 + |CD|2 = |AC |2

42 + |CD|2 = 112

|CD|2 = 105

|CD| =√

105

|CD| ≈ 10.247

Using the formula A = 12bh, the area of ∆ABC is

A = 12 × 8×

√105 ≈ 40.988 square units.

J. Garvin — The Pythagorean Theorem

Slide 8/10

tr i gonometry

Pythagorean Theorem

Example

Verify that ∆JKL contains a right angle.

If ∆JKL contains a right angle,the Pythagorean Theorem willhold true.

The sum of the squares of thearms is 282 + 452 = 2 809.

The square of the hypotenuse is532 = 2 809.

Since we obtain the same value,∠L = 90◦, as it is across from thehypotenuse.

J. Garvin — The Pythagorean Theorem

Slide 9/10

tr i gonometry

Pythagorean Theorem

Example

Verify that ∆JKL contains a right angle.

If ∆JKL contains a right angle,the Pythagorean Theorem willhold true.

The sum of the squares of thearms is 282 + 452 = 2 809.

The square of the hypotenuse is532 = 2 809.

Since we obtain the same value,∠L = 90◦, as it is across from thehypotenuse.

J. Garvin — The Pythagorean Theorem

Slide 9/10

tr i gonometry

Pythagorean Theorem

Example

Verify that ∆JKL contains a right angle.

If ∆JKL contains a right angle,the Pythagorean Theorem willhold true.

The sum of the squares of thearms is 282 + 452 = 2 809.

The square of the hypotenuse is532 = 2 809.

Since we obtain the same value,∠L = 90◦, as it is across from thehypotenuse.

J. Garvin — The Pythagorean Theorem

Slide 9/10

tr i gonometry

Pythagorean Theorem

Example

Verify that ∆JKL contains a right angle.

If ∆JKL contains a right angle,the Pythagorean Theorem willhold true.

The sum of the squares of thearms is 282 + 452 = 2 809.

The square of the hypotenuse is532 = 2 809.

Since we obtain the same value,∠L = 90◦, as it is across from thehypotenuse.

J. Garvin — The Pythagorean Theorem

Slide 9/10

tr i gonometry

Pythagorean Theorem

Example

Verify that ∆JKL contains a right angle.

If ∆JKL contains a right angle,the Pythagorean Theorem willhold true.

The sum of the squares of thearms is 282 + 452 = 2 809.

The square of the hypotenuse is532 = 2 809.

Since we obtain the same value,∠L = 90◦, as it is across from thehypotenuse.

J. Garvin — The Pythagorean Theorem

Slide 9/10

tr i gonometry

Questions?

J. Garvin — The Pythagorean Theorem

Slide 10/10