Question

1 use the diagram to match each description to the appropriate equation
\\(\frac{ab}{ac} = \frac{ac}{ad}\\) \\(\frac{ab}{cb} = \frac{cb}{db}\\) \\(\frac{ad}{cd} = \frac{cd}{db}\\)
the length of the altitude is the geometric mean of the hypotenuse segments
the length of the short leg is the geometric mean of the length of the hypotenuse and the length of the short segment
the length of the long leg is the geometric mean of the length of the hypotenuse and the length of the long segment
2 given \\(jkl\\) jamie writes a proportion to find \\(mj\\)
\\(\frac{12}{18} = \frac{18}{12 + mj}\\)
jamie’s proportion is incorrect fill in the blanks to write the correct proportion
\\(\frac{12}{18} = \frac{12}{18}\\) (note: original has a handwritten 12 and 18, might be typo but ocr as is)

1. fill in the blanks and solve the proportion to find \\(qp\\)

\\(\frac{\quad}{qp} = \frac{qp}{\quad}\\) (first blank and second blank)
\\(\frac{\quad}{qp} = \frac{qp}{\quad}\\) (repeat structure, maybe typo)
\\((qp)^2 = \\)
\\(qp = \\)

Explanation:

Step1: Match altitude-geometric mean

The altitude to hypotenuse of a right triangle is the geometric mean of the two hypotenuse segments. This corresponds to $\frac{AD}{CD} = \frac{CD}{DB}$.

Step2: Match short leg-geometric mean

The short leg is the geometric mean of the hypotenuse and the adjacent hypotenuse segment. This corresponds to $\frac{AB}{AC} = \frac{AC}{AD}$.

Step3: Match long leg-geometric mean

The long leg is the geometric mean of the hypotenuse and the adjacent hypotenuse segment. This corresponds to $\frac{AB}{CB} = \frac{CB}{DB}$.

Step4: Correct Jamie's proportion

For right triangle altitude rule: $\frac{\text{Short segment}}{\text{Leg}} = \frac{\text{Leg}}{\text{Whole hypotenuse}}$. Here, $\frac{12}{MJ} = \frac{MJ}{12+18}$, so $\frac{12}{18} = \frac{12}{MJ}$.

Step5: Set up QP proportion

For right triangle short leg rule: $\frac{\text{Hypotenuse}}{\text{Short leg}} = \frac{\text{Short leg}}{\text{Adjacent segment}}$.
$\frac{4+28}{QP} = \frac{QP}{4}$

Step6: Cross-multiply to solve

Cross-multiply to find $(QP)^2$:
$(QP)^2 = 4 \times (4+28) = 4 \times 32 = 128$

Step7: Calculate QP

Take square root:
$QP = \sqrt{128} = 8\sqrt{2}$

Answer:

1.

• The length of the altitude is the geometric mean of the hypotenuse segments: $\boldsymbol{\frac{AD}{CD} = \frac{CD}{DB}}$
• The length of the short leg is the geometric mean of the length of the hypotenuse and the length of the short segment: $\boldsymbol{\frac{AB}{AC} = \frac{AC}{AD}}$
• The length of the long leg is the geometric mean of the length of the hypotenuse and the length of the long segment: $\boldsymbol{\frac{AB}{CB} = \frac{CB}{DB}}$

2.
$\boldsymbol{\frac{12}{18} = \frac{12}{MJ}}$

3.
$\frac{32}{QP} = \frac{QP}{4}$
$(QP)^2 = 128$
$\boldsymbol{QP = 8\sqrt{2}}$