. A right-angled triangle is to be constructed so that the hypotenuse is 20 cm long. The height of the
triangle is 4 cm longer than the base of the triangle. Evaluate the length of each side of the triangle.
[Verify your answer using MATHEMATICA]

To solve this problem, let's define the lengths of the sides of the triangle and use the Pythagorean theorem.

### Step 1: Define the variables
- Let \( x \) represent the length of the base of the right-angled triangle.
- The height of the triangle is 4 cm longer than the base, so it can be represented as \( x + 4 \).
- The hypotenuse is given as 20 cm.

### Step 2: Apply the Pythagorean theorem
According to the Pythagorean theorem:
\[
\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2
\]
Substitute the values:
\[
20^2 = x^2 + (x + 4)^2
\]
This simplifies to:
\[
400 = x^2 + (x^2 + 8x + 16)
\]
\[
400 = 2x^2 + 8x + 16
\]
### Step 3: Solve the quadratic equation
Now, rearrange the equation:
\[
2x^2 + 8x + 16 - 400 = 0
\]
\[
2x^2 + 8x - 384 = 0
\]
Divide by 2 to simplify:
\[
x^2 + 4x - 192 = 0
\]
Solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( x^2 + 4x - 192 = 0 \), \( a = 1 \), \( b = 4 \), and \( c = -192 \).

Substitute the values into the formula:
\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-192)}}{2(1)}
\]
\[
x = \frac{-4 \pm \sqrt{16 + 768}}{2}
\]
\[
x = \frac{-4 \pm \sqrt{784}}{2}
\]
\[
x = \frac{-4 \pm 28}{2}
\]
Now, solve for the two possible values of \( x \):
\[
x = \frac{-4 + 28}{2} = \frac{24}{2} = 12
\]
or
\[
x = \frac{-4 - 28}{2} = \frac{-32}{2} = -16
\]
Since a length cannot be negative, the base of the triangle is \( x = 12 \) cm.

### Step 4: Find the height
The height is 4 cm longer than the base:
\[
\text{Height} = x + 4 = 12 + 4 = 16 \text{ cm}.
\]

### Step 5: Verify using the Pythagorean theorem
Let's verify the result by checking the Pythagorean theorem:
\[
\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2
\]
\[
20^2 = 12^2 + 16^2
\]
\[
400 = 144 + 256
\]
\[
400 = 400
\]
The equation holds true, so the solution is correct.

### Final Answer:
- The base of the triangle is 12 cm.
- The height of the triangle is 16 cm.
- The hypotenuse is 20 cm.

Do you need the Mathematica verification? And would you like more details or have any questions?  

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Here are 5 related questions:
1. What is the relationship between the sides of a right-angled triangle?
2. How would the formula change if the height was 5 cm shorter than the base?
3. What happens if the hypotenuse is a decimal, say 15.5 cm?
4. How do we derive the quadratic formula used in this solution?
5. Can we find the area of this triangle, and how?

**Tip:** In a right-angled triangle, the Pythagorean theorem is key for solving unknown sides if you know any two sides.

A right-angled triangle is to be constructed so that the hypotenuse is 20 cm long. The height of the triangle is 4 cm longer than the base of the triangle. Evaluate the length of each side of the triangle.

Learn how to solve a right-angled triangle problem where the hypotenuse is 20 cm, and the height is 4 cm longer than the base. Follow a detailed step-by-step solution using the Pythagorean theorem and quadratic equations. Suitable for students in Grades 9-10.

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