Question

model it
complete the problems about the pythagorean theorem.
here is another way to think about the pythagorean theorem:
in a right triangle, the area of a square drawn on the hypotenuse is equal to the combined areas of the squares drawn on the two legs.
you can see an example of this in the diagram to the right.
a. write the length of each side of the right triangle.
a: ______ units
b: ______ units
c: ______ units
b. write the area of each square.
square with side length a: ______ square units
square with side length b: ______ square units
square with side length c: ______ square units
c. use your answers from problems 3a and 3b to show that this diagram supports the pythagorean theorem. identify which sides of the right triangle are the legs and which is the hypotenuse in your answer.
discuss it
ask: how does a square with side length a represent a²?

Explanation:

Step1: Count side - lengths

By counting the grid - squares along each side of the right - triangle, we find that \(a = 3\) units, \(b = 4\) units, and \(c = 5\) units.

Step2: Calculate square areas

The area of a square is given by \(A=s^{2}\), where \(s\) is the side - length. So the area of the square with side - length \(a\) is \(a^{2}=3^{2}=9\) square units, the area of the square with side - length \(b\) is \(b^{2}=4^{2}=16\) square units, and the area of the square with side - length \(c\) is \(c^{2}=5^{2}=25\) square units.

Step3: Verify Pythagorean Theorem

The legs of the right - triangle are \(a\) and \(b\), and the hypotenuse is \(c\). The combined areas of the squares on the legs are \(a^{2}+b^{2}=9 + 16=25\) square units, and the area of the square on the hypotenuse is \(c^{2}=25\) square units. Since \(a^{2}+b^{2}=c^{2}\), the diagram supports the Pythagorean Theorem.

Answer:

a. \(a:3\) units, \(b:4\) units, \(c:5\) units
b. Square with side - length \(a:9\) square units, Square with side - length \(b:16\) square units, Square with side - length \(c:25\) square units
c. The legs of the right - triangle are \(a\) and \(b\), and the hypotenuse is \(c\). We have \(a^{2}=9\), \(b^{2}=16\), and \(c^{2}=25\). Since \(a^{2}+b^{2}=9 + 16=25=c^{2}\), the diagram supports the Pythagorean Theorem.