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.