Question

1. what is the area of the large sized box?

area = 25 in²
12 in
area =?

1. what is the side length of the medium sized box?

15 cm
area=81 cm²
?

1. what is the length of the hypotenuse?

area=49 m²
area = 576 m²

1. what is the area of the small sized box?

20 un
area=?
area = 256 un²

1. what is the side length of the small sized box?

area = 625 cm²
?
20 cm

1. what is the area of the triangle?

6 in
area = 100 in²
~exit ticket~
in the space at right, draw a right

Explanation:

Step1: Apply Pythagorean theorem (areas)

The area of the large square equals the sum of the areas of the two smaller squares.
$\text{Area}_{\text{large}} = 25 + 12^2 = 25 + 144 = 169$

Step2: Find side length of small square

First, get the side length from its area: $\sqrt{81} = 9$. Then use Pythagorean theorem for the triangle's sides: the unknown side $s$ satisfies $s^2 + 9^2 = 15^2$.
$s^2 = 225 - 81 = 144$, so $s = \sqrt{144} = 12$

Step3: Calculate hypotenuse length

First, find the side lengths of the two squares: $\sqrt{49} = 7$, $\sqrt{576} = 24$. The hypotenuse $c$ satisfies $c^2 = 7^2 + 24^2$.
$c^2 = 49 + 576 = 625$, so $c = \sqrt{625} = 25$

Step4: Find small square area

Let the area of the small square be $A$. The side of the large square is 20, so $20^2 = A + 256$.
$A = 400 - 256 = 144$

Step5: Find small square side length

First, find the side of the large square: $\sqrt{625} = 25$. Let the small square side be $s$, so $25^2 = s^2 + 20^2$.
$s^2 = 625 - 400 = 225$, so $s = \sqrt{225} = 15$

Step6: Calculate triangle area

First, find the other leg of the right triangle: the large square has side $\sqrt{100} = 10$. Let the unknown leg be $b$, so $6^2 + b^2 = 10^2$.
$b^2 = 100 - 36 = 64$, so $b = 8$. Area of triangle is $\frac{1}{2} \times 6 \times 8 = 24$

Answer:

1. $169$ in²
2. $12$ cm
3. $25$ m
4. $144$ un²
5. $15$ cm
6. $24$ in²