Question

analyze the diagram below and complete the instructions that follow.
find the value of each of the variables.
a. ( x = 12, y = 15, z = 20 )
b. ( x = 9, y = 15, z = 20 )
c. ( x = 9, y = 18, z = 20 )
d. ( x = 12, y = 18, z = 20 )

Explanation:

Step1: Recall geometric mean theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. Also, each leg is the geometric mean of the hypotenuse and the adjacent segment. Let the right triangle have hypotenuse \( c \), legs \( a,b \), and altitude \( h \) to hypotenuse, dividing hypotenuse into segments \( m,n \). Then \( h^2 = m\times n \), \( a^2 = c\times m \), \( b^2 = c\times n \).

Here, altitude is \( 12 \), one segment of hypotenuse is \( x \), another is \( 16 \)? Wait, no, the right triangle has legs \( y \) and \( z \), hypotenuse \( x + 16 \)? Wait, no, looking at the diagram: there is a right triangle with a perpendicular from the right angle to the hypotenuse, creating two smaller right triangles. Let's denote: the large right triangle has legs \( y \) and \( z \), hypotenuse \( L=x + 16 \)? Wait, no, the segments on the hypotenuse: one segment is \( x \), the other is \( 16 \), and the altitude is \( 12 \). Also, the leg adjacent to segment \( x \) is \( y \), and adjacent to \( 16 \) is \( z \)? Wait, no, let's correct:

In the right triangle, when we draw an altitude from the right angle to the hypotenuse, we have three similar triangles: the large triangle, and two smaller ones. So, the altitude (12) is the geometric mean of the two segments of the hypotenuse: \( 12^2=x\times16 \)? Wait, no, \( 12^2 = x\times16 \)? Wait, \( 12^2 = x\times16 \) would be \( 144 = 16x \), \( x = 9 \). Then, the leg \( y \) (adjacent to segment \( x \)) is the geometric mean of hypotenuse segment \( x \) and the whole hypotenuse? Wait, no, the leg \( y \) is the geometric mean of the hypotenuse segment \( x \) and the sum of \( x \) and \( 16 \)? Wait, no, the formula is: if the hypotenuse is divided into segments \( m \) and \( n \), then leg \( a \) (opposite to segment \( n \)) has \( a^2 = m(m + n) \)? No, correct formula: for a right triangle with hypotenuse \( c \), divided into \( m \) and \( n \) by altitude \( h \), then \( h^2 = m\times n \), \( a^2 = m\times c \), \( b^2 = n\times c \), where \( a \) and \( b \) are legs.

So here, let \( m = x \), \( n = 16 \), \( h = 12 \). Then \( h^2 = m\times n \implies 12^2 = x\times16 \implies 144 = 16x \implies x = 9 \). Then, the leg \( y \) (which is adjacent to \( x \)): \( y^2 = x\times(x + 16) \)? Wait, no, \( y^2 = x\times(x + 16) \)? Wait, no, the hypotenuse of the large triangle is \( x + 16 \), and the leg \( y \) is the geometric mean of \( x \) and \( (x + 16) \)? Wait, no, actually, the leg \( y \) is the geometric mean of the hypotenuse segment \( x \) and the entire hypotenuse? No, the correct formula is: if the hypotenuse is \( c = m + n \), then \( a^2 = m\times c \), \( b^2 = n\times c \). Wait, no, let's take the two smaller triangles: the triangle with legs \( x \) and \( 12 \) is similar to the triangle with legs \( 12 \) and \( 16 \), and to the large triangle.

So, triangle with legs \( x \) and \( 12 \) is similar to triangle with legs \( 12 \) and \( 16 \). So, the ratio of corresponding sides: \( \frac{x}{12}=\frac{12}{16} \implies 16x = 144 \implies x = 9 \). Then, the leg \( y \) (the other leg of the large triangle) can be found by Pythagoras in the small triangle: \( y^2 = x^2 + 12^2 = 9^2 + 12^2 = 81 + 144 = 225 \implies y = 15 \). Then, the leg \( z \): in the other small triangle, \( z^2 = 16^2 + 12^2 = 256 + 144 = 400 \implies z = 20 \). So \( x = 9 \), \( y = 15 \), \( z = 20 \), which is option B.

Step1: Calculate \( x \) using geometric mean of altitude

The altitude (12) to the hypotenuse is the geometric mean of the two segments (\( x \) and 16) of the hypotenuse. So:
$$12^2 = x \times 16$$
$$144 = 16x$$
$$x = \frac{144}{16} = 9$$

Step2: Calculate \( y \) using Pythagoras in small triangle

The leg \( y \) is the hypotenuse of the small triangle with legs \( x = 9 \) and altitude \( 12 \). So:
$$y^2 = 9^2 + 12^2$$
$$y^2 = 81 + 144 = 225$$
$$y = \sqrt{225} = 15$$

Step3: Calculate \( z \) using Pythagoras in other small triangle

The leg \( z \) is the hypotenuse of the small triangle with legs \( 16 \) and altitude \( 12 \). So:
$$z^2 = 16^2 + 12^2$$
$$z^2 = 256 + 144 = 400$$
$$z = \sqrt{400} = 20$$

Answer:

B. \( x = 9,y = 15,z = 20 \)