Triangle WXM is shown, where mangle X=90^circ ,XM=3sqrt (10) and WX=7 What is the ratio for tanM A (3sqrt (10))/(sqrt (139)) B (7)/(sqrt (139)) C (7)/(3sqrt (10)) D (3sqrt (10))/(7)

To find the ratio for \(\tan M\), we need to know the lengths of the sides adjacent to and opposite angle \(M\) in triangle \(WXM\). Since \(\angle X\) is a right angle, we can use the Pythagorean theorem to find the length of \(WM\), the hypotenuse of the right triangle \(WXM\).The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In triangle \(WXM\), \(WX\) and \(XM\) are the two sides that form the right angle, and \(WM\) is the hypotenuse.Given:\(WX = 7\)\(XM = 3\sqrt{10}\)Step 1: Apply the Pythagorean theorem to find \(WM\).\[WM^2 = WX^2 + XM^2\]\[WM^2 = 7^2 + (3\sqrt{10})^2\]\[WM^2 = 49 + 9 \cdot 10\]\[WM^2 = 49 + 90\]\[WM^2 = 139\]\[WM = \sqrt{139}\]Step 2: Find \(\tan M\).The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. For \(\tan M\), \(XM\) is the opposite side and \(WX\) is the adjacent side.\[\tan M = \frac{XM}{WX}\]\[\tan M = \frac{3\sqrt{10}}{7}\]The final answer is:(D) \(\frac{3 \sqrt{10}}{7}\)