Question

segment.

1. given $\triangle jkl$, 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.
$\frac{12}{18} = \underline{\quad}$

1. fill in the blanks and solve the proportion to

find $qp$.
$\frac{\quad}{qp} = \frac{qp}{\quad}$
$\frac{\quad}{qp} = \frac{qp}{\quad}$
$(qp)^2 = $
$qp = $

Explanation:

Problem 2

Step1: Identify similar triangles

In $\triangle JKL$, $\triangle KML \sim \triangle KLJ$ (right triangles share $\angle K$), so corresponding sides are proportional. The short leg of $\triangle KML$ is 12, short leg of $\triangle KLJ$ is $KM + MJ = 12 + MJ$; the long leg of $\triangle KML$ is 18, long leg of $\triangle KLJ$ is $JL$, but the correct proportion matches short leg to long leg of each triangle: $\frac{\text{Short leg of } \triangle KML}{\text{Long leg of } \triangle KML} = \frac{\text{Short leg of } \triangle KLJ}{\text{Long leg of } \triangle KLJ}$.

Step2: Write correct proportion

$\frac{12}{18} = \frac{18}{12 + MJ}$

Step1: Identify geometric mean theorem

In right $\triangle QPR$, $QP$ is the altitude to hypotenuse $QR$, so by geometric mean theorem: $\frac{QS}{QP} = \frac{QP}{QR}$.

Step2: Substitute known values

$QS = 4$, $QR = 4 + 28 = 32$, so $\frac{4}{QP} = \frac{QP}{32}$

Step3: Cross-multiply to solve

$(QP)^2 = 4 \times 32 = 128$

Step4: Calculate square root

$QP = \sqrt{128} = 8\sqrt{2}$

Answer:

$\boldsymbol{\frac{18}{12 + MJ}}$

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Problem 3