Question

directions: find the value of x. 1 2. 3. 4.

Explanation:

Step1: Apply Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\).

Step2: Solve for \(x\) in the first triangle

Given legs \(a = 7\) and \(b = 10\), then \(x^{2}=7^{2}+10^{2}=49 + 100=149\), so \(x=\sqrt{149}\).

Step3: Solve for \(x\) in the second triangle

Given hypotenuse \(c = 21\) and one leg \(a = 19\), then \(x^{2}=21^{2}-19^{2}=(21 + 19)(21 - 19)=40\times2 = 80\), so \(x=\sqrt{80}=4\sqrt{5}\).

Step4: Solve for \(x\) in the third triangle

Given legs \(a = 16\) and \(b = 27\), then \(x^{2}=16^{2}+27^{2}=256+729 = 985\), so \(x=\sqrt{985}\).

Step5: Solve for \(x\) in the fourth triangle

Since the triangle is isosceles right - triangle with height \(h = 18\) and hypotenuse \(x\), and for an isosceles right - triangle \(x=\sqrt{2}h\), so \(x = 18\sqrt{2}\).

Answer:

1. \(x=\sqrt{149}\)
2. \(x = 4\sqrt{5}\)
3. \(x=\sqrt{985}\)
4. \(x = 18\sqrt{2}\)