Question

7.1 ap-6
two dimensions of a right triangle are 5 units and 13 units. a student writes the equation $5^2 + 13^2 = c^2$ to find the length of the third side. answer parts a and b.
a. if all the side lengths are integers, is the students equation correct? explain.
a. yes. $5^2 + 13^2 = 25 + 169 = 194$. $sqrt{194}$ is 14, which is an integer.
b. no. $5^2 + 13^2 = 15 + 189 = 204$. $sqrt{204}$ is 14.3, which is not an integer.
c. no. $5^2 + 13^2 = 25 + 169 = 194$. $sqrt{194}$ is about 13.9, which is not an integer.
d. yes. $5^2 + 13^2 = 25 + 299 = 324$. $sqrt{324}$ is 18, which is an integer.
click to select your answer and then click check answer.
1 part remaining

Explanation:

Step1: Verify Pythagorean setup

The student assumes $5^2 + 13^2 = c^2$, but first calculate the sum:
$5^2 + 13^2 = 25 + 169 = 194$

Step2: Check integer validity

Calculate $\sqrt{194} \approx 13.9$, which is not an integer. Since all side lengths are integers, the student's equation is incorrect because it does not yield an integer third side (and also fails to account for 13 possibly being the hypotenuse, but the core issue is the non-integer result here).

Answer:

C. No. $5^2 + 13^2 = 25 + 169 = 194$. $\sqrt{194}$ is about 13.9, which is not an integer.