Question

given the figure shown below, determine the length of c.
draw
36 in²
25 in²
c
show your work here
hint: to add the square root symbol (√□), type
oot\

Explanation:

Step1: Find the legs of the right triangle

The area of the first square is \(36\space\text{in}^2\), so the side length (let's call it \(a\)) is \(\sqrt{36} = 6\space\text{in}\). The area of the second square is \(25\space\text{in}^2\), so the side length (let's call it \(b\)) is \(\sqrt{25}=5\space\text{in}\).

Step2: Apply the Pythagorean theorem

For a right triangle, \(c^2=a^2 + b^2\). Substituting \(a = 6\) and \(b = 5\), we get \(c^2=36 + 25=61\). Then \(c=\sqrt{61}\space\text{in}\) (wait, no, wait. Wait, the square on the hypotenuse: wait, the first square is on one leg, area \(36\), so leg \(a = 6\), second square area \(25\), leg \(b = 5\). Then the square on the hypotenuse \(c\) has area \(a^2 + b^2=36 + 25 = 61\)? Wait, no, wait the figure: the blue triangle is a right triangle, with legs as sides of the squares with areas 36 and 25. So the hypotenuse is \(c\), and the square on the hypotenuse has area \(c^2\), which should be equal to the sum of the areas of the squares on the legs (Pythagorean theorem: in a right triangle, the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides). So \(c^2=36 + 25=61\)? Wait, no, wait 36 is \(6^2\), 25 is \(5^2\), so \(c^2=6^2 + 5^2=36 + 25 = 61\), so \(c=\sqrt{61}\)? Wait, but maybe I made a mistake. Wait, no, the Pythagorean theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), \(a^2 + b^2=c^2\). So if the area of the square on \(a\) is \(a^2 = 36\), area on \(b\) is \(b^2 = 25\), then area on \(c\) is \(c^2=a^2 + b^2=36 + 25 = 61\), so \(c=\sqrt{61}\)? Wait, but 6 and 5: 6-5-√61? Wait, no, 6 and 5: 6²+5²=36+25=61, so yes. Wait, but maybe the figure is a right triangle with legs 6 and 5, so hypotenuse c, and the square on c has area 61, so c is √61. Wait, but let me check again.

Wait, the first square: area 36, so side length is √36=6. Second square: area 25, side length √25=5. The triangle is right-angled, so by Pythagoras, hypotenuse c: \(c^2=6^2 + 5^2=36 + 25=61\), so \(c=\sqrt{61}\) inches. Wait, but maybe I misread the figure. Wait, the square on the hypotenuse is the one with side c, so its area is \(c^2\), which is equal to the sum of the areas of the other two squares (36 and 25). So \(c^2=36 + 25=61\), so \(c=\sqrt{61}\).

Answer:

\(\sqrt{61}\) inches (or approximately 7.81 inches, but exact form is \(\sqrt{61}\))