Question

pythagorean theorem
find the length of the missing side in the following examples. round answers to the nearest hundredth (2 decimal places), if necessary. show all work
1.
2.
3.
4.
5.
6.
7.
8.
9.

Explanation:

1. For the first triangle with sides 3cm and 6cm:

• # Explanation:
• ## Step1: Recall Pythagorean theorem
• For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). Here \(a = 3\) and \(b = 6\), and we want to find the hypotenuse \(x\).
• \(x=\sqrt{3^{2}+6^{2}}\)
• ## Step2: Calculate the values
• First, \(3^{2}=9\) and \(6^{2}=36\). Then \(3^{2}+6^{2}=9 + 36=45\). So \(x=\sqrt{45}\approx6.71\) cm.
• # Answer:
• \(x\approx6.71\) cm

1. For the second triangle with sides 4cm and 7cm:

• # Explanation:
• ## Step1: Apply Pythagorean theorem
• We want to find the hypotenuse \(x\), so \(x=\sqrt{4^{2}+7^{2}}\).
• \(x=\sqrt{16 + 49}\)
• ## Step2: Compute the sum
• \(16+49 = 65\), so \(x=\sqrt{65}\approx8.06\) cm.
• # Answer:
• \(x\approx8.06\) cm

1. For the third triangle with sides 14cm and 22cm:

• # Explanation:
• ## Step1: Use Pythagorean theorem
• \(x=\sqrt{14^{2}+22^{2}}\).
• \(x=\sqrt{196+484}\)
• ## Step2: Calculate the result
• \(196 + 484=680\), so \(x=\sqrt{680}\approx26.08\) cm.
• # Answer:
• \(x\approx26.08\) cm

1. For the fourth triangle with hypotenuse 4.2cm and one leg 3.7cm:

• # Explanation:
• ## Step1: Rearrange Pythagorean theorem
• If \(c = 4.2\) and \(a = 3.7\), then \(b=\sqrt{c^{2}-a^{2}}\), so \(x=\sqrt{4.2^{2}-3.7^{2}}\).
• First, \(4.2^{2}=17.64\) and \(3.7^{2}=13.69\). Then \(4.2^{2}-3.7^{2}=17.64 - 13.69 = 3.95\).
• \(x=\sqrt{3.95}\approx1.99\) cm.
• # Answer:
• \(x\approx1.99\) cm

1. For the fifth triangle with sides 5.3m and 6.9m:

• # Explanation:
• ## Step1: Apply Pythagorean theorem
• \(x=\sqrt{5.3^{2}+6.9^{2}}\).
• \(x=\sqrt{28.09+47.61}\)
• ## Step2: Find the value of \(x\)
• \(28.09+47.61 = 75.7\), so \(x=\sqrt{75.7}\approx8.70\) m.
• # Answer:
• \(x\approx8.70\) m

1. For the sixth triangle with hypotenuse 8.6cm and one leg 2.7cm:

• # Explanation:
• ## Step1: Rearrange Pythagorean theorem
• \(x=\sqrt{8.6^{2}-2.7^{2}}\).
• \(8.6^{2}=73.96\) and \(2.7^{2}=7.29\). Then \(8.6^{2}-2.7^{2}=73.96 - 7.29 = 66.67\).
• \(x=\sqrt{66.67}\approx8.17\) cm.
• # Answer:
• \(x\approx8.17\) cm

1. For the seventh triangle with sides 17m and 22m:

• # Explanation:
• ## Step1: Use Pythagorean theorem
• \(x=\sqrt{17^{2}+22^{2}}\).
• \(x=\sqrt{289+484}\)
• ## Step2: Compute \(x\)
• \(289+484 = 773\), so \(x=\sqrt{773}\approx27.80\) m.
• # Answer:
• \(x\approx27.80\) m

1. For the eighth triangle with hypotenuse 6.9cm and one leg 6.4cm:

• # Explanation:
• ## Step1: Rearrange Pythagorean theorem
• \(x=\sqrt{6.9^{2}-6.4^{2}}\).
• \(6.9^{2}=47.61\) and \(6.4^{2}=40.96\). Then \(6.9^{2}-6.4^{2}=47.61 - 40.96 = 6.65\).
• \(x=\sqrt{6.65}\approx2.58\) cm.
• # Answer:
• \(x\approx2.58\) cm

1. For the ninth triangle with sides 3m and 7m:

• # Explanation:
• ## Step1: Apply Pythagorean theorem
• \(x=\sqrt{3^{2}+7^{2}}\).
• \(x=\sqrt{9 + 49}\)
• ## Step2: Calculate \(x\)
• \(9+49 = 58\), so \(x=\sqrt{58}\approx7.62\) m.
• # Answer:
• \(x\approx7.62\) m

Answer:

1. For the first triangle with sides 3cm and 6cm:

• # Explanation:
• ## Step1: Recall Pythagorean theorem
• For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). Here \(a = 3\) and \(b = 6\), and we want to find the hypotenuse \(x\).
• \(x=\sqrt{3^{2}+6^{2}}\)
• ## Step2: Calculate the values
• First, \(3^{2}=9\) and \(6^{2}=36\). Then \(3^{2}+6^{2}=9 + 36=45\). So \(x=\sqrt{45}\approx6.71\) cm.
• # Answer:
• \(x\approx6.71\) cm

1. For the second triangle with sides 4cm and 7cm:

• # Explanation:
• ## Step1: Apply Pythagorean theorem
• We want to find the hypotenuse \(x\), so \(x=\sqrt{4^{2}+7^{2}}\).
• \(x=\sqrt{16 + 49}\)
• ## Step2: Compute the sum
• \(16+49 = 65\), so \(x=\sqrt{65}\approx8.06\) cm.
• # Answer:
• \(x\approx8.06\) cm

1. For the third triangle with sides 14cm and 22cm:

• # Explanation:
• ## Step1: Use Pythagorean theorem
• \(x=\sqrt{14^{2}+22^{2}}\).
• \(x=\sqrt{196+484}\)
• ## Step2: Calculate the result
• \(196 + 484=680\), so \(x=\sqrt{680}\approx26.08\) cm.
• # Answer:
• \(x\approx26.08\) cm

1. For the fourth triangle with hypotenuse 4.2cm and one leg 3.7cm:

• # Explanation:
• ## Step1: Rearrange Pythagorean theorem
• If \(c = 4.2\) and \(a = 3.7\), then \(b=\sqrt{c^{2}-a^{2}}\), so \(x=\sqrt{4.2^{2}-3.7^{2}}\).
• First, \(4.2^{2}=17.64\) and \(3.7^{2}=13.69\). Then \(4.2^{2}-3.7^{2}=17.64 - 13.69 = 3.95\).
• \(x=\sqrt{3.95}\approx1.99\) cm.
• # Answer:
• \(x\approx1.99\) cm

1. For the fifth triangle with sides 5.3m and 6.9m:

• # Explanation:
• ## Step1: Apply Pythagorean theorem
• \(x=\sqrt{5.3^{2}+6.9^{2}}\).
• \(x=\sqrt{28.09+47.61}\)
• ## Step2: Find the value of \(x\)
• \(28.09+47.61 = 75.7\), so \(x=\sqrt{75.7}\approx8.70\) m.
• # Answer:
• \(x\approx8.70\) m

1. For the sixth triangle with hypotenuse 8.6cm and one leg 2.7cm:

• # Explanation:
• ## Step1: Rearrange Pythagorean theorem
• \(x=\sqrt{8.6^{2}-2.7^{2}}\).
• \(8.6^{2}=73.96\) and \(2.7^{2}=7.29\). Then \(8.6^{2}-2.7^{2}=73.96 - 7.29 = 66.67\).
• \(x=\sqrt{66.67}\approx8.17\) cm.
• # Answer:
• \(x\approx8.17\) cm

1. For the seventh triangle with sides 17m and 22m:

• # Explanation:
• ## Step1: Use Pythagorean theorem
• \(x=\sqrt{17^{2}+22^{2}}\).
• \(x=\sqrt{289+484}\)
• ## Step2: Compute \(x\)
• \(289+484 = 773\), so \(x=\sqrt{773}\approx27.80\) m.
• # Answer:
• \(x\approx27.80\) m

1. For the eighth triangle with hypotenuse 6.9cm and one leg 6.4cm:

• # Explanation:
• ## Step1: Rearrange Pythagorean theorem
• \(x=\sqrt{6.9^{2}-6.4^{2}}\).
• \(6.9^{2}=47.61\) and \(6.4^{2}=40.96\). Then \(6.9^{2}-6.4^{2}=47.61 - 40.96 = 6.65\).
• \(x=\sqrt{6.65}\approx2.58\) cm.
• # Answer:
• \(x\approx2.58\) cm

1. For the ninth triangle with sides 3m and 7m:

• # Explanation:
• ## Step1: Apply Pythagorean theorem
• \(x=\sqrt{3^{2}+7^{2}}\).
• \(x=\sqrt{9 + 49}\)
• ## Step2: Calculate \(x\)
• \(9+49 = 58\), so \(x=\sqrt{58}\approx7.62\) m.
• # Answer:
• \(x\approx7.62\) m