Question

find the value of x in these diagrams.
3.
(diagram of a rectangle with length 10, width 5, and a diagonal labeled x)

Explanation:

Step1: Identify the triangle type

The diagram shows a right - triangle (since the rectangle has right angles and the diagonal forms a right - triangle with legs 5 and 10). We can use the Pythagorean theorem, which states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(a = 5\), \(b = 10\) and \(c=x\).

Step2: Apply the Pythagorean theorem

Substitute the values of \(a\) and \(b\) into the formula: \(x^{2}=5^{2}+10^{2}\)
Calculate \(5^{2}=25\) and \(10^{2}=100\), so \(x^{2}=25 + 100=125\)

Step3: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{125}\)
Simplify \(\sqrt{125}=\sqrt{25\times5}=5\sqrt{5}\approx5\times2.236 = 11.18\) (if we want a decimal approximation) or leave it in radical form \(5\sqrt{5}\).

Answer:

If in radical form, \(x = 5\sqrt{5}\); if as a decimal approximation, \(x\approx11.18\) (usually, the exact form \(5\sqrt{5}\) or the approximate decimal form is acceptable depending on the requirements. If we calculate \(5\sqrt{5}\), \(\sqrt{5}\approx2.236\), so \(5\times2.236 = 11.18\))