Q5:

The altitude of a right triangle is $7$ cm less than its base. If the hypotenuse is $13$ cm, find the other two sides.

Given:

• A right-angled triangle.
• The length of the hypotenuse is $13$ cm.
• The altitude (height) is $7$ cm less than the base.

To find:

• The lengths of the base and the altitude of the triangle.

Step 1: Defining Variables

Let the base of the right triangle be $x$ cm.

According to the problem, the altitude is $7$ cm less than the base.

Therefore, the altitude = $(x - 7)$ cm.

Step 2: Applying the Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagorean Theorem):

$(\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Altitude})^2$

Substituting the given values:

$13^2 = x^2 + (x - 7)^2$

Step 3: Expanding and Simplifying the Equation

$169 = x^2 + (x^2 - 14x + 49)$ [Using the identity $(a - b)^2 = a^2 - 2ab + b^2$]

$169 = 2x^2 - 14x + 49$

Subtract $169$ from both sides to set the quadratic equation to zero:

$0 = 2x^2 - 14x + 49 - 169$

$2x^2 - 14x - 120 = 0$

Divide the entire equation by $2$ to simplify:

$x^2 - 7x - 60 = 0$

Step 4: Solving the Quadratic Equation by Factorization

We need to find two numbers that multiply to $-60$ and add to $-7$. These numbers are $-12$ and $5$.

$x^2 - 12x + 5x - 60 = 0$

$x(x - 12) + 5(x - 12) = 0$

$(x - 12)(x + 5) = 0$

Step 5: Determining the Possible Values for x

Setting each factor to zero:

1) $x - 12 = 0 \implies x = 12$

2) $x + 5 = 0 \implies x = -5$

Since the length of a side of a triangle cannot be negative, we discard $x = -5$.

Therefore, the base $x = 12$ cm.

Step 6: Calculating the Altitude

Altitude = $x - 7$

Altitude = $12 - 7 = 5$ cm.

Final Answer: The base of the triangle is 12 cm and the altitude is 5 cm.