To find the length of the third side for each triangle, we use the Pythagorean theorem. This rule applies to right-angled triangles (triangles with a square corner). The formula is:
$$a^2 + b^2 = c^2$$
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$a$ and
$b$ are the two shorter sides (legs) that touch the right angle.
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$c$ is the longest side (hypotenuse), which is opposite the right angle.
Here is the step-by-step solution for each triangle shown in the worksheet.
Triangle 1 (Top Left)
We need to find the hypotenuse ($c$). The legs are 17 and 32.
1. Square the lengths of the legs:
$$17^2 = 289$$
$$32^2 = 1024$$
2. Add them together:
$$289 + 1024 = 1313$$
3. Take the square root of the sum to find $c$:
$$c = \sqrt{1313}$$
$$c \approx 36.24$$
Triangle 2 (Top Right)
We need to find a leg ($a$). The other leg is 35, and the hypotenuse is 47. Since we are looking for a leg, we subtract the square of the known leg from the square of the hypotenuse.
1. Set up the equation:
$$a^2 + 35^2 = 47^2$$
2. Square the known numbers:
$$a^2 + 1225 = 2209$$
3. Subtract 1225 from 2209:
$$a^2 = 2209 - 1225$$
$$a^2 = 984$$
4. Take the square root:
$$a = \sqrt{984}$$
$$a \approx 31.37$$
Triangle 3 (Bottom Left)
We need to find the hypotenuse ($c$). The legs are 80 and 85.
1. Square the lengths of the legs:
$$80^2 = 6400$$
$$85^2 = 7225$$
2. Add them together:
$$6400 + 7225 = 13625$$
3. Take the square root of the sum to find $c$:
$$c = \sqrt{13625}$$
$$c \approx 116.73$$
Triangle 4 (Bottom Right)
We need to find a leg ($a$). The other leg is 46, and the hypotenuse is 49.
1. Set up the equation:
$$a^2 + 46^2 = 49^2$$
2. Square the known numbers:
$$a^2 + 2116 = 2401$$
3. Subtract 2116 from 2401:
$$a^2 = 2401 - 2116$$
$$a^2 = 285$$
4. Take the square root:
$$a = \sqrt{285}$$
$$a \approx 16.88$$
Final Answer:
The lengths of the third sides are approximately:
1.
36.24
2.
31.37
3.
116.73
4.
16.88
Parent Tip: Review the logic above to help your child master the concept of worksheet pythagorean theorem.
