Question

using algebra find the value of each variable.
27.
(there is a right triangle with an altitude drawn to the hypotenuse, the segment of the hypotenuse adjacent to the side of length 7 is labeled 7, the altitude is labeled 5, and the hypotenuse is labeled m)

Explanation:

Step1: Identify triangle area formula

The area of a triangle can be calculated as $\frac{1}{2} \times \text{base} \times \text{height}$. Here, we can also recognize that the triangle is composed of two smaller right triangles, and we can use the geometric mean (altitude-on-hypotenuse) theorem for right triangles. For a right triangle, the length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments it divides the hypotenuse into. Let the unknown segment of the hypotenuse be $n$, so $m = 7 + n$. First, find $n$:
$$5^2 = 7 \times n$$

Step2: Solve for unknown segment $n$

Rearrange the equation to solve for $n$:
$$n = \frac{25}{7}$$

Step3: Calculate total hypotenuse $m$

Add the two segments of the hypotenuse:
$$m = 7 + \frac{25}{7} = \frac{49}{7} + \frac{25}{7} = \frac{74}{7}$$

Answer:

$m = \frac{74}{7}$