Question

ex. 1) what variable represents the hypotenuse?
b) if p = 8 and r = 15 then w = ____.
ex. 2) what variable represents the hypotenuse?
b) if p = 25 and r = 24 then w = ____.

1. calculate the missing side length (x) in each triangle to 1 decimal place (as needed).

a)
b)
c)
d)

Explanation:

Step1: Identify the triangle type (right triangle)

All triangles are right - angled, so we use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse and \(a,b\) are the legs.

Part (a)

Step2: Apply Pythagorean theorem

Here, \(a = 6\), \(b = 8\), and \(x\) is the hypotenuse. So \(x^{2}=6^{2}+8^{2}\)
\(x^{2}=36 + 64=100\)
\(x=\sqrt{100}=10\)

Part (b)

Step2: Apply Pythagorean theorem

Here, \(a = 9\), \(b = 15\), and \(x\) is the hypotenuse. So \(x^{2}=9^{2}+15^{2}\)
\(x^{2}=81+225 = 306\)
\(x=\sqrt{306}\approx17.5\) (rounded to 1 decimal place)

Part (c)

Step2: Apply Pythagorean theorem (solve for leg)

Here, \(c = 20\), \(a = 12\), and \(x\) is a leg. So \(x^{2}=c^{2}-a^{2}\)
\(x^{2}=20^{2}-12^{2}=400 - 144 = 256\)
\(x=\sqrt{256}=16\)

Part (d)

Step2: Apply Pythagorean theorem (solve for leg)

Here, \(c = 34\), \(a = 16\), and \(x\) is a leg. So \(x^{2}=c^{2}-a^{2}\)
\(x^{2}=34^{2}-16^{2}=1156 - 256 = 900\)
\(x=\sqrt{900}=30\)

Answer:

a) \(x = 10\)
b) \(x\approx17.5\)
c) \(x = 16\)
d) \(x = 30\)