Question

carls work
step 1: find the vertex of a possible right triangle. v(4, 1)
step 2: find the leg lengths of the right triangle. nv = 7, mv = 5
step 3: substitute the values into the pythagorean theorem. 7² + 5² = c²
step 4: solve for c.
7² + 5² = c²
49 + 25 = c²
74 = c²
√74 = c
which best describes the accuracy of carls solution?
carls answer is correct, but he incorrectly labeled the leg lengths in step 2.
carls work and answer are both entirely correct.
carls work and answer are incorrect. he substituted and solved incorrectly.
carls work and answer are incorrect. he chose a vertex, v, that does not make a right triangle with mn.

Explanation:

Step1: Recall the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\), where \(a\) and \(b\) are the legs of a right - triangle and \(c\) is the hypotenuse. For a right - triangle, the two legs are perpendicular to each other.

Step2: Analyze the vertex \(V(4,1)\)

If we assume \(N\) and \(M\) are two other points, to form a right - triangle with \(V\), we need to check the perpendicularity. The lengths \(NV = 7\) and \(MV=5\) are just the distances from \(V\) to \(N\) and \(V\) to \(M\). But we don't know if \(\angle NV M\) is a right angle. In fact, if we consider the general coordinate - based right - triangle condition (using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), and for two perpendicular lines \(m_1\times m_2=- 1\)), we find that Carl wrongly assumed that \(NV\) and \(MV\) are the legs of a right - triangle. He chose a vertex \(V\) that may not form a right - triangle with \(MN\).

Answer:

Carl’s work and answer are incorrect. He chose a vertex, \(V\), that does not make a right triangle with \(MN\).