Question

1. a) calculate the areas of the three squares. b) is the triangle a right triangle? explain. image shows a yellow triangle with sides labeled 4 cm, 3 cm, and ?, with three gray squares attached to its sides (side lengths 4 cm, 3 cm, 2 cm).

Explanation:

Part (a)

To find the area of a square, we use the formula \( A = s^2 \), where \( s \) is the side length of the square.

Step 1: Area of the square with side 4 cm

The side length \( s = 4 \) cm. Using the area formula for a square:
\( A_1 = 4^2 = 16 \) \( \text{cm}^2 \)

Step 2: Area of the square with side 3 cm

The side length \( s = 3 \) cm. Using the area formula for a square:
\( A_2 = 3^2 = 9 \) \( \text{cm}^2 \)

Step 3: Area of the square with side 2 cm

The side length \( s = 2 \) cm. Using the area formula for a square:
\( A_3 = 2^2 = 4 \) \( \text{cm}^2 \)

Part (b)

To determine if a triangle is a right triangle, we use the Pythagorean theorem, which states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), \( a^2 + b^2 = c^2 \). Here, the sides of the triangle are 2 cm, 3 cm, and 4 cm. Let's check the Pythagorean theorem:

• Let \( a = 2 \), \( b = 3 \), \( c = 4 \)
• Calculate \( a^2 + b^2 \): \( 2^2 + 3^2 = 4 + 9 = 13 \)
• Calculate \( c^2 \): \( 4^2 = 16 \)

Since \( 13
eq 16 \), the Pythagorean theorem does not hold. Thus, the triangle is not a right triangle.

Answer:

s:

(a)

• Area of the square with side 4 cm: \( \boldsymbol{16 \text{ cm}^2} \)
• Area of the square with side 3 cm: \( \boldsymbol{9 \text{ cm}^2} \)
• Area of the square with side 2 cm: \( \boldsymbol{4 \text{ cm}^2} \)

(b)

The triangle is not a right triangle. Explanation: For a right triangle, the sum of the squares of the two shorter sides (legs) should equal the square of the longest side (hypotenuse). Here, \( 2^2 + 3^2 = 4 + 9 = 13 \), and \( 4^2 = 16 \). Since \( 13
eq 16 \), the Pythagorean theorem is not satisfied, so the triangle is not a right triangle.