MPM2D: Principles of Mathematicsjongarvin.com/up/MPM2D/slides/pythagorean_theorem.pdf ·...
Transcript of MPM2D: Principles of Mathematicsjongarvin.com/up/MPM2D/slides/pythagorean_theorem.pdf ·...
tr i gonometry
MPM2D: Principles of Mathematics
The Pythagorean Theorem
J. Garvin
Slide 1/10
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
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
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 |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
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
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