Question

question 11
find the missing side in each triangle using any method. check your answers using a different method.
(there are two right triangles: the first has legs 3 and x, hypotenuse 5; the second has legs 9 and 12, hypotenuse y. input boxes for x and y with units label.)

Explanation:

Step1: Find x in the small triangle

The small triangle is a right triangle with hypotenuse 5 and one leg 3. Using the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c = 5\) and \(b = 3\), we solve for \(a\) (which is \(x\)):

$$ x^2 + 3^2 = 5^2 \\ x^2 + 9 = 25 \\ x^2 = 25 - 9 \\ x^2 = 16 \\ x = \sqrt{16} \\ x = 4 $$

Step2: Determine the scale factor

The two triangles are similar (both right triangles with the same acute angle marked). The legs of the small triangle are 3 and 4, and the corresponding leg of the large triangle is 9 (since 3 corresponds to 9). The scale factor is \(\frac{9}{3}=3\).

Step3: Find y in the large triangle

The hypotenuse of the small triangle is 5, so the hypotenuse \(y\) of the large triangle is \(5\times3 = 15\). We can also verify using the Pythagorean theorem for the large triangle with legs 9 and 12:

$$ 9^2 + 12^2 = y^2 \\ 81 + 144 = y^2 \\ 225 = y^2 \\ y = \sqrt{225} \\ y = 15 $$

Answer:

\(x = 4\) units, \(y = 15\) units