In a right-angled triangle ΔDEF, let ∠D = 90° and let EF be the hypotenuse with length 12 cm. If DX is the median drawn from vertex D to the hypotenuse EF, what is the length (in cm) of DX?

Introduction / Context:
This question tests a special property of medians in right angled triangles. In any right triangle, the median drawn from the right angle vertex to the hypotenuse has a fixed relationship with the hypotenuse. Knowing this property allows you to answer without needing the other side lengths.

Given Data / Assumptions:
• Triangle DEF is right angled at D, so ∠D = 90°.
• EF is the hypotenuse and its length is 12 cm.
• DX is the median from D to hypotenuse EF, so X is the midpoint of EF.
• We must find the length DX.

Concept / Approach:
In a right angled triangle, the midpoint of the hypotenuse is equidistant from all three vertices. This means that the median from the right angle vertex to the midpoint of the hypotenuse is equal to half the hypotenuse and also equal to the circumradius of the triangle. As a result, the length of that median is simply half the hypotenuse length.

Step-by-Step Solution:
Step 1: Recognize the property: in any right triangle, the median from the right angle to the hypotenuse equals half of the hypotenuse. Step 2: Here EF is the hypotenuse and EF = 12 cm. Step 3: The midpoint X of EF satisfies EX = XF = EF / 2 = 12 / 2 = 6 cm. Step 4: The distance DX from the right angle vertex to X is also equal to 6 cm by the special property of the right triangle median.

Verification / Alternative check:
You can verify the property by placing a right triangle with coordinates. For example, take D at (0, 0), E at (a, 0), and F at (0, b). The hypotenuse EF has midpoint X at (a/2, b/2). Distance DX equals √[(a/2)2 + (b/2)2] = (1/2)√(a2 + b2), while EF = √(a2 + b2). Thus DX = EF / 2, confirming the property.

Why Other Options Are Wrong:
• 3 cm or 4 cm would imply that the median is less than half the hypotenuse, contradicting the geometric property derived from coordinates.
• 12 cm would make the median equal to the entire hypotenuse length, which is impossible because the median joins the right angle vertex to the midpoint of the hypotenuse, not to its end.

Common Pitfalls:
A frequent error is to attempt to use Pythagoras theorem and introduce extra unknowns for the legs, which is unnecessary. Remember the special properties of right triangles, especially for medians and circumcircles, to quickly answer questions like this.

Final Answer:
The length of DX is 6 cm.