Question

(a) find the length of each side of the right triangle. distance between (5, 0) and (10, 0) $d_1 = $blank distance between (10, 0) and (10, 12) $d_2 = $blank distance between (5, 0) and (10, 12) $d_3 = $blank (b) show that these lengths satisfy the pythagorean theorem. $d_1^{2}=$blank $d_2^{2}=$blank $d_1^{2}+d_2^{2}=$blank=$d_3^{2}$

Explanation:

Step1: Calculate $d_1$

Use the distance formula for two - points $(x_1,y_1)$ and $(x_2,y_2)$ which is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $(5,0)$ and $(10,0)$, $x_1 = 5,y_1=0,x_2 = 10,y_2 = 0$. Then $d_1=\sqrt{(10 - 5)^2+(0 - 0)^2}=\sqrt{5^2}=5$.

Step2: Calculate $d_2$

For points $(10,0)$ and $(10,12)$, $x_1 = 10,y_1 = 0,x_2=10,y_2 = 12$. Then $d_2=\sqrt{(10 - 10)^2+(12 - 0)^2}=\sqrt{0 + 12^2}=12$.

Step3: Calculate $d_3$

For points $(5,0)$ and $(10,12)$, $x_1 = 5,y_1 = 0,x_2 = 10,y_2 = 12$. Then $d_3=\sqrt{(10 - 5)^2+(12 - 0)^2}=\sqrt{5^2+12^2}=\sqrt{25 + 144}=\sqrt{169}=13$.

Step4: Check Pythagorean theorem

Calculate $d_1^2$, $d_2^2$ and $d_1^2 + d_2^2$. $d_1^2=5^2 = 25$, $d_2^2=12^2=144$, $d_1^2 + d_2^2=25+144 = 169$. And $d_3^2=13^2 = 169$. So $d_1^2 + d_2^2=d_3^2$.

Answer:

(a) $d_1 = 5$, $d_2 = 12$, $d_3 = 13$
(b) $d_1^2=25$, $d_2^2 = 144$, $d_1^2 + d_2^2=169$, $d_3^2=169$