Construct a right triangle whose base is 12 cm and sum of its hypotenuse and other side is 18 cm.

To construct a right triangle with the given conditions, we solve it step by step. Here's the detailed explanation:

### Given:
1. Base of the triangle, \( b = 12 \) cm
2. Sum of hypotenuse (\( h \)) and the other side (\( p \)) = \( 18 \) cm

### Step 1: Relationship between hypotenuse, base, and perpendicular
In a right triangle, by the Pythagorean theorem:
\[ h^2 = b^2 + p^2 \]

Substituting \( b = 12 \):
\[ h^2 = 12^2 + p^2 \]
\[ h^2 = 144 + p^2 \] (1)

### Step 2: Relationship between hypotenuse and the other side
It is given that:
\[ h + p = 18 \]
From this, express \( h \) in terms of \( p \):
\[ h = 18 - p \] (2)

### Step 3: Substitution
Substitute \( h = 18 - p \) into equation (1):
\[ (18 - p)^2 = 144 + p^2 \]
Expand \( (18 - p)^2 \):
\[ 324 - 36p + p^2 = 144 + p^2 \]

Cancel \( p^2 \) from both sides:
\[ 324 - 36p = 144 \]

Simplify:
\[ 180 = 36p \]

Solve for \( p \):
\[ p = \frac{180}{36} = 5 \] cm

### Step 4: Find hypotenuse
From equation (2):
\[ h = 18 - p \]
\[ h = 18 - 5 = 13 \] cm

### Step 5: Verify
Check using the Pythagorean theorem:
\[ h^2 = b^2 + p^2 \]
\[ 13^2 = 12^2 + 5^2 \]
\[ 169 = 144 + 25 \]
\[ 169 = 169 \] (Correct)

### Final Answer:
The right triangle has:
- Base = 12 cm
- Perpendicular = 5 cm
- Hypotenuse = 13 cm