In a `Delta PQR , angle Q = 90^(@) , PQ = 5 ,QR = 12` find QM if M is centroid

Answer

To find the length of QM in triangle PQR where angle Q is 90 degrees, PQ = 5, and QR = 12, and M is the centroid, we can follow these steps:

### Step 1: Identify the triangle and its sides
We have triangle PQR with:
- Angle Q = 90 degrees
- PQ = 5 (one leg)
- QR = 12 (the other leg)

### Step 2: Use the Pythagorean theorem to find PR
In a right triangle, the Pythagorean theorem states:
\[ PR^2 = PQ^2 + QR^2 \]

Substituting the values:
\[ PR^2 = 5^2 + 12^2 \]
\[ PR^2 = 25 + 144 \]
\[ PR^2 = 169 \]
\[ PR = \sqrt{169} = 13 \]

### Step 3: Determine the coordinates of the points
Assuming:
- Point Q is at the origin (0, 0)
- Point P is at (0, 5) since PQ is vertical
- Point R is at (12, 0) since QR is horizontal

So, the coordinates are:
- Q(0, 0)
- P(0, 5)
- R(12, 0)

### Step 4: Find the coordinates of the centroid M
The centroid M of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) is given by:
\[ M\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]

Substituting the coordinates of points P, Q, and R:
\[ M\left(\frac{0 + 0 + 12}{3}, \frac{5 + 0 + 0}{3}\right) \]
\[ M\left(\frac{12}{3}, \frac{5}{3}\right) \]
\[ M(4, \frac{5}{3}) \]

### Step 5: Calculate the length of QM
To find the length QM, we can use the distance formula:
\[ QM = \sqrt{(x_M - x_Q)^2 + (y_M - y_Q)^2} \]

Substituting the coordinates of M and Q:
\[ QM = \sqrt{(4 - 0)^2 + \left(\frac{5}{3} - 0\right)^2} \]
\[ QM = \sqrt{4^2 + \left(\frac{5}{3}\right)^2} \]
\[ QM = \sqrt{16 + \frac{25}{9}} \]
To combine these, convert 16 to a fraction with a denominator of 9:
\[ QM = \sqrt{\frac{144}{9} + \frac{25}{9}} \]
\[ QM = \sqrt{\frac{169}{9}} \]
\[ QM = \frac{\sqrt{169}}{\sqrt{9}} \]
\[ QM = \frac{13}{3} \]

### Final Answer
Thus, the length of QM is \( \frac{13}{3} \).

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