What is the measure of the hypotenuse of a right triangle, when its medians, drawn from the vertices of the acute angles, are 5 cm and `2 sqrt(10) cm`

Options

A. `5 sqrt(8) cm`
B. `2 sqrt(13) cm`
C. `6 sqrt(10) cm`
D. `2 sqrt(7) cm`

Answer

To find the measure of the hypotenuse of a right triangle when the medians from the acute angles are given, we can follow these steps:

### Step 1: Define the triangle and medians
Let triangle ABC be a right triangle where angle C is the right angle. The lengths of the sides opposite to vertices A, B, and C are denoted as AB = X, BC = Y, and AC = Z (the hypotenuse). The medians from vertices A and B are given as AE = 5 cm and CF = 2√10 cm.

**Hint:** Remember that the median from a vertex to the midpoint of the opposite side divides the triangle into two smaller triangles.

### Step 2: Apply the formula for the median
The formula for the length of the median from a vertex to the opposite side in a triangle is given by:
\[ \text{Median} = \frac{1}{2} \sqrt{2a^2 + 2b^2 - c^2} \]
where a and b are the lengths of the sides adjacent to the vertex, and c is the length of the opposite side.

### Step 3: Set up equations for the medians
1. For median AE (from A to BC):
\[ AE^2 = \frac{2AB^2 + 2AC^2 - BC^2}{4} \]
Substituting AE = 5 cm:
\[ 5^2 = \frac{2X^2 + 2Z^2 - Y^2}{4} \]
Simplifying gives:
\[ 25 = \frac{2X^2 + 2Z^2 - Y^2}{4} \implies 100 = 2X^2 + 2Z^2 - Y^2 \quad \text{(Equation 1)} \]

2. For median CF (from B to AC):
\[ CF^2 = \frac{2BC^2 + 2BA^2 - AC^2}{4} \]
Substituting CF = 2√10:
\[ (2\sqrt{10})^2 = \frac{2Y^2 + 2X^2 - Z^2}{4} \]
Simplifying gives:
\[ 40 = \frac{2Y^2 + 2X^2 - Z^2}{4} \implies 160 = 2Y^2 + 2X^2 - Z^2 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( 100 = 2X^2 + 2Z^2 - Y^2 \)
2. \( 160 = 2Y^2 + 2X^2 - Z^2 \)

We can add these two equations to eliminate one variable:
\[ 100 + 160 = (2X^2 + 2Z^2 - Y^2) + (2Y^2 + 2X^2 - Z^2) \]
This simplifies to:
\[ 260 = 4X^2 + Y^2 + Z^2 \]

### Step 5: Use Pythagorean theorem
Since triangle ABC is a right triangle:
\[ Z^2 = X^2 + Y^2 \]
Substituting this into the equation gives:
\[ 260 = 4X^2 + (Z^2 - X^2) \implies 260 = 3X^2 + Z^2 \]

### Step 6: Substitute and solve for Z
From \(Z^2 = X^2 + Y^2\) and substituting \(Y^2\) from the first equation, we can find:
\[ Z^2 = 52 \implies Z = \sqrt{52} = 2\sqrt{13} \]

### Conclusion
Thus, the measure of the hypotenuse Z is:
\[ \text{Hypotenuse} = 2\sqrt{13} \text{ cm} \]

### Final Answer
The measure of the hypotenuse is \(2\sqrt{13} \text{ cm}\).

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