Question

1. if the area of square 1 is 289 units² and the area of square 3 is 64 units², what is the area of square 2? explain your reasoning.
2. denise is using a ladder to clean the outside of her second - story windows. the ladder she is using is 34 feet long, and she puts the base of the ladder 12 feet away from the house in order to avoid her flower garden. how high up the side of her house does the ladder reach? round to the nearest tenth if necessary.
3. will ran the diagonal distance across a square field measuring 40 yards on each side. james ran the diagonal distance across a rectangular field with a length of 30 yards and a width of 35 yards. who ran a longer distance, and how much longer did he run? show work to justify your answer.

Explanation:

Problem 9

Step1: Find side lengths of squares

Let side of square 1: $s_1 = \sqrt{289} = 17$
Let side of square 3: $s_3 = \sqrt{169} = 13$

Step2: Use Pythagorean theorem

Square 2 side $s_2$: $s_2^2 = s_1^2 - s_3^2$
$ s_2^2 = 17^2 - 13^2 = 289 - 169 = 120$

Step3: Area of square 2 is $s_2^2$

Area = $120$

Step1: Define Pythagorean variables

Ladder (hypotenuse) $c=24$ ft, base $a=13$ ft, height $b=?$

Step2: Rearrange Pythagorean theorem

$b = \sqrt{c^2 - a^2}$

Step3: Calculate height

$b = \sqrt{24^2 - 13^2} = \sqrt{576 - 169} = \sqrt{407} \approx 20.2$

Step1: Calculate Will's distance

Square diagonal: $d_W = 40\sqrt{2} \approx 56.57$ yards

Step2: Calculate James's distance

Rectangle diagonal: $d_J = \sqrt{25^2 + 35^2} = \sqrt{625 + 1225} = \sqrt{1850} \approx 43.01$ yards

Step3: Compare and find difference

$56.57 > 43.01$, Difference: $56.57 - 43.01 = 13.56$

Answer:

$120$ units$^2$

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Problem 10