E2.4 Describe the Pythagorean relationship using various geometric models, and apply the theorem to solve problems involving an unknown side length for a given right triangle.
Skill: Explaining the Pythagorian Theorem Using Various Geometric Models
The properties of a right triangle can be used to find an unknown side length. The longest side of a right triangle is always opposite the 90° angle and it is called the hypotenuse. Given any right triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the other two sides. This is known as the Pythagorean relationship. The length of the hypotenuse squared and the length of each of the sides squared can be represented as three squares formed with the three sides of the triangle, and then visualizing the sum of the areas of the two smaller squares as being equivalent to the area of the larger square.
Image a small white square labeled: "one" is in the center of a large square. The rest of the surface of this square is divided into 4 rectangular triangles, they are all marked: 6. The side of this square is "c". In the lower right corner a second square, of medium size, is placed, it is marked 16. A third square of small size, and marked 9, is placed in the lower left corner. In this way the three squares create a triangular shaped space, "a" "b" "c". Area of the large square equals "c" squared. Area of the middle square equals "b" squared. Area of small square equals "a" squared. Area of "c" squared equals "a" squared plus "b" sqared.
The Pythagorean theorem expresses this relationship symbolically: a2 + b2 = c2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides of the triangle. For example:
- if side a is 3 units long, then a square constructed on this side has an area of 32 or 9 square units;
- if side b is 4 units long, then a square constructed on this side has an area of 42 or 16 square units;
- if the square on side c is equal to the combined areas of the squares on sides a and b, then the square on side c must have an area of 25 square units (9 square units + 16 square units);
- if the area of the square on side c is 25 square units, then the length of c must be \(\sqrt{25}\) or 5 units.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
In this kind of problem, the most common error is not correctly identifying the longest side, the hypotenuse.
Skill: Applying the Theorem to Calculate the Missing Length of One Side of a Given Right Triangle
The inverse relationship between addition and subtraction means that the Pythagorean theorem can be used to find any length on a right triangle (for example, c2 − b2 = a2; c2 − a2 = b2).
The Pythagorean theorem is used to indirectly measure lengths that would be impractical or impossible to measure directly. For example, the theorem is used extensively in construction, architecture, and navigation, and extensions of the theorem are used to measure distances in space.
Note
The properties of a square can be used to find its side length or area. The side length of a square is equal to the square root of its area (see also B1.3).
Image A square of 2 units squared, labeled: “square root 2 units squared.” A square of 5 units squared labeled: “5 units squared.” A square of ten units squared labeled: “ ten units squared.”
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Example
Determine the missing measurement. Round your answer to the nearest tenth.
Visual Representation
This is a right triangle. I know that for any right triangle, the Pythagorean relationship implies that the area of a square constructed on its longest side (the hypotenuse) is equal to the combined areas of the squares constructed on the two sides adjacent to the right angle.
Since I need to find the length of side b, I need to subtract the area of the square of side a from the area of the square of the hypotenuse.
Image A square with a vertex on the right of segment "a" touches the top of segment "h" of a second square. The bottom of segment "h" of this second square touches the right of a segment named "b" of a third square. This last one touches the left of the segment "a" of the first square. The squares form a triangle "a" "h" "b". The angle "a" "b" is a right angle. The area of the first square is "a" squared. The area of "a" squared equals 16 multiplied by 16, equals 256 square centimeters. The area of the second square equals "h" squared. The area of the second square is "h" squared equals 24 multiplied by 24, equals 576 square centimeters. The area of the third square equals "b" squared. The area of the third square equals "b" squared equals 576 minus 256, equals 320 square centimeters.
The length of side b is the square root of \(320 \mathrm{cm}^{2}\).
\(\sqrt{320} \ \mathrm{cm}^{2} \approx 17,9 \ \mathrm{cm}\)
The missing measurement, the length of side b, is about 17.9 cm.
Symbolic Representation
This is a right triangle, so I use the Pythagorean theorem to determine the missing measurement:
\(b^{2} + 16^{2} = 24^{2}\)
\(b^{2} + 256 \ \mathrm{cm}^{2} = 576 \ \mathrm{cm}^{2}\)
\(b^{2} + 256 \mathrm{cm}^{2} \ - \ 256 \ \mathrm{cm}^{2} = 576 \ \ \ \mathrm{cm}^{2} \ - \ 256 \ \mathrm{cm}^{2} \)
\(b^{2} = 320 \mathrm{cm}^{2}\)
\(b = \sqrt{320} \mathrm{cm}^{2}\)
\(b \approx 17,9 \ \mathrm{cm}\)
The missing measurement, the base of the right triangle, has a length of approximately 17.9 cm.
Source: translated from En avant, les maths!, 8e année, CM, Sens de l’espace, p. 5-6.
Knowledge : Pythagorean Relationship
Given any right triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the other two sides.
\(a^{2} + b^{2} = c^{2}\)
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.