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Math  /  Geometry

QuestionTwo congruent squares are shown in Figures 1 and 2 below.
Figure 1
Figure 2 se the drop-down menus to complete the proof of the Pythagorean Theorem using the figures. lick the arrows to choose an answer from each menu.
The combined area of the shaded triangles in Figure 1 is Choose... the combined area of the shaded triangles in Figure 2. The area of the unshaded square in Figure 1 can be represented by Choose... □\square - The combined area of the two unshaded squares in Figure 2 can be represented by Choose... . The areas of the squares in Figure 1 and Figure ress

Studdy Solution

STEP 1

1. Both figures are based on the same large square with side length a+ba+b.
2. The figures are used to prove the Pythagorean Theorem, which states that in a right triangle with legs aa and bb and hypotenuse cc, the relationship a2+b2=c2a^2 + b^2 = c^2 holds.
3. The triangles in both figures are congruent right triangles.

STEP 2

1. Calculate the combined area of the shaded triangles in both figures.
2. Determine the area of the unshaded square in Figure 1.
3. Determine the combined area of the two unshaded squares in Figure 2.
4. Relate the areas to complete the proof of the Pythagorean Theorem.

STEP 3

Calculate the area of one right triangle with legs aa and bb. The area of one triangle is given by:
Area of one triangle=12×a×b \text{Area of one triangle} = \frac{1}{2} \times a \times b
Since there are four triangles in each figure, the combined area of the shaded triangles in each figure is:
Combined area of shaded triangles=4×12×a×b=2ab \text{Combined area of shaded triangles} = 4 \times \frac{1}{2} \times a \times b = 2ab
The combined area of the shaded triangles in Figure 1 is equal to the combined area of the shaded triangles in Figure 2.

STEP 4

The area of the unshaded square in Figure 1 is the area of the smaller square with side length cc:
Area of unshaded square in Figure 1=c2 \text{Area of unshaded square in Figure 1} = c^2

STEP 5

In Figure 2, the large square is divided into two smaller squares with side lengths aa and bb. The combined area of these two unshaded squares is:
Combined area of unshaded squares in Figure 2=a2+b2 \text{Combined area of unshaded squares in Figure 2} = a^2 + b^2

STEP 6

Since the total area of the large square in both figures is the same, we equate the expressions derived from each figure:
(a+b)2=c2+2ab (a+b)^2 = c^2 + 2ab (from Figure 1)
(a+b)2=a2+b2+2ab (a+b)^2 = a^2 + b^2 + 2ab (from Figure 2)
Since both expressions represent the total area of the large square, we equate the unshaded areas:
c2=a2+b2 c^2 = a^2 + b^2
This completes the proof of the Pythagorean Theorem.
The proof of the Pythagorean Theorem using the figures is completed.

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