Question

Study the diagram. Then complete each statement.

Short answer

1. The final answer is $2$ and $7.$

Thus $z=14$

Step-by-step solution

Step 1. Given information:

A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

Step 2. Concept use.

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below), the relation between the hypotenuse and the normal to it is expressed as,

$h^2=xy$

$h=\sqrt{xy}$

This implies the length of the normal is the geometric mean of the two parts of the hypotenuse divided by it.

Step 3.  Applying the concept.    

Consider the triangle shown below.

For the given triangle, the altitude is represented by $z$and the parts of the hypotenuse are $2$and $7$.

Thus, comparing with the general figure and formula, the length \[z\]can be calculated as,

$z=\sqrt{2\times 7}$

$=\sqrt{14}$

Therefore, the geometric mean of $z$ is $2$ and $7$.

b. The final answer is $2$and $9.$

Thus $x=3\sqrt{2}$

Step 1. Given information:

A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

Step 2. Concept used.

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below), the both small triangles are congruent to each other and to the bigger triangle. This gives the relation between the hypotenuse of the bigger triangle and one of the legs of smaller triangle as,

$b^2=cx$

$b=\sqrt{cx}$

Step 3.  Applying the concept.    

Consider the triangle shown below.

For the given triangle, the one of the legs of the bigger triangle is represented by $x$ and the parts of the hypotenuse considered is $2$ and the total length of hypotenuse being $9.$

Thus, comparing with the general figure and formula, the value of $u$ can be calculated as,

$x=\sqrt{2\times 9}$

$=\sqrt{18}$

$=3\sqrt{2}$

Therefore, the geometric mean of $s$ is $2$and $9.$

a. The final answer is $7$ and $9.$

Thus $y=3\sqrt{7}$

Step 1. Given information:

A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

Step 2. Concept used.

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below), the both small triangles are congruent to each other and to the bigger triangle. This gives the relation between the hypotenuse of the bigger triangle and one of the legs of smaller triangle as,

$a^2=cy$

$a=\sqrt{cy}$

Step 3.  Applying the concept.    

Consider the triangle shown below.

For the given triangle, the one of the legs of the bigger triangle is represented by $y$ and the parts of the hypotenuse considered is $7$ and the total length of hypotenuse being $9.$

Thus, comparing with the general figure and formula, the value of $y$ can be calculated as,

$y=\sqrt{7\times 9}$

$=\sqrt{63}$

$=3\sqrt{7}$

Therefore, the geometric mean of $y$ is $7$ and $9.$