Question

the pythagorean theorem
right triangles (triangles with right - angles) have a very special property. the areas of squares built on the two shorter sides add up to the area of the square built on the longest side. the longest side in every right triangle is the only one which is not a side of the right - angle. it is called the hypotenuse.
in symbols, we write
if a, b and c are sides of a right triangle and c is the hypotenuse, then
$a^{2}+b^{2}=c^{2}$.
this is called the pythagorean theorem after the greek mathematician, pythagoras. you can check that it works for the right triangle above. (16 + 9 = 25)
the pythagorean theorem enables us to find the length of the third side of a right triangle when we know the lengths of the other two sides. all we have to do is use the formula.
find the length of the third side of each triangle. if the length is not a rational number, round it off to the nearest hundredth.
a = 12
b = 5
$a = 5$
$b = 12$
$5^{2}+12^{2}=c^{2}$
$25 + 144 = c^{2}$
$169 = c^{2}$
$c = 13$ or $c=-13$
-13 doesnt make sense as a length.

Explanation:

Step1: Recall Pythagorean theorem

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

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

Given \(a = 12\), \(b = 5\), then \(c=\sqrt{a^{2}+b^{2}}=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\).

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

Given \(a = 35\), \(b = 35\), then \(c=\sqrt{35^{2}+35^{2}}=\sqrt{2\times35^{2}} = 35\sqrt{2}\approx49.50\).

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

Given \(a = 6\), \(b = 8\), then \(c=\sqrt{6^{2}+8^{2}}=\sqrt{36+64}=\sqrt{100}=10\).

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

Given \(a = 15\), \(b = 9\), then \(c=\sqrt{15^{2}+9^{2}}=\sqrt{225 + 81}=\sqrt{306}\approx17.49\).

Answer:

1. For the triangle with \(a = 12\) and \(b = 5\), \(c = 13\)
2. For the triangle with two equal - length sides \(a=b = 35\), \(c=\sqrt{35^{2}+35^{2}}=\sqrt{2\times35^{2}}=35\sqrt{2}\approx49.50\)
3. For the triangle with \(a = 6\) and \(b = 8\), \(c=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)
4. For the triangle with \(a = 15\) and \(b = 9\), \(c=\sqrt{15^{2}+9^{2}}=\sqrt{225+81}=\sqrt{306}\approx17.49\)