Question

use the pythagorean theorem to solve for the unknown length of a leg (a or b) or the unknown length of the hypotenuse c, the unknown with its appropriate value.

a = 2, b = 3 and c = ?

a = ?, b = 4 and c = 6

a = 7, b = ?, c = 10

a = 5, b = 8 and c = ?

drag & drop the answer

√13
√51
√89
2√5

Explanation:

First Problem: \(a = 2, b = 3\) and \(c = ?\)

Step 1: Recall Pythagorean theorem

The Pythagorean theorem states that for a right triangle, \(a^{2}+b^{2}=c^{2}\). Substitute \(a = 2\) and \(b = 3\) into the formula.
\(c^{2}=2^{2}+3^{2}\)

Step 2: Calculate the values

\(2^{2}=4\) and \(3^{2}=9\), so \(c^{2}=4 + 9=13\). Then take the square root of both sides: \(c=\sqrt{13}\)

Step 1: Recall Pythagorean theorem and rearrange for \(a\)

From \(a^{2}+b^{2}=c^{2}\), we can solve for \(a\) as \(a^{2}=c^{2}-b^{2}\). Substitute \(b = 4\) and \(c = 6\) into the formula.
\(a^{2}=6^{2}-4^{2}\)

Step 2: Calculate the values

\(6^{2}=36\) and \(4^{2}=16\), so \(a^{2}=36-16 = 20\). Then \(a=\sqrt{20}=2\sqrt{5}\) (since \(\sqrt{20}=\sqrt{4\times5}=2\sqrt{5}\))

Step 1: Recall Pythagorean theorem and rearrange for \(b\)

From \(a^{2}+b^{2}=c^{2}\), we can solve for \(b\) as \(b^{2}=c^{2}-a^{2}\). Substitute \(a = 7\) and \(c = 10\) into the formula.
\(b^{2}=10^{2}-7^{2}\)

Step 2: Calculate the values

\(10^{2}=100\) and \(7^{2}=49\), so \(b^{2}=100 - 49=51\). Then \(b=\sqrt{51}\)

Answer:

\(\sqrt{13}\)

Second Problem: \(a = ?, b = 4\) and \(c = 6\)