Question

estimate the length of the diagonal.
______ cm
calculate the length of the diagonal.
______ cm

Explanation:

Estimate the length of the diagonal:

Step1: Recall square diagonal properties

A square with side length \( s \) has a diagonal \( d \) related by the Pythagorean theorem (\( d^{2}=s^{2}+s^{2} \)). For \( s = 1 \), \( d^{2}=2 \), so \( d=\sqrt{2}\approx1.414 \). Visually, the diagonal is longer than the side (1), and since \( 1^{2}+1^{2}=2 \), and \( 1.4^{2}=1.96 \), \( 1.5^{2}=2.25 \), so estimate ~1.4.

Step1: Apply Pythagorean theorem

For a square with side length \( s = 1 \), the diagonal \( d \) forms a right triangle with two sides. By \( a^{2}+b^{2}=c^{2} \) (here \( a = b = 1 \), \( c = d \)):
\( d^{2}=1^{2}+1^{2}=1 + 1 = 2 \)

Step2: Solve for \( d \)

Take square root: \( d=\sqrt{2}\approx1.414 \) (exact value is \( \sqrt{2} \), approximate decimal is ~1.414)

Answer:

\( 1.4 \) (approximate, reasonable estimates like 1.4 - 1.5 are acceptable)

Calculate the length of the diagonal: