Given that the triangles \( \triangle JKL \sim \triangle PKQ \), we can use the properties of similar triangles to find the lengths of the sides of triangle JKL based on the lengths of triangle PKQ.
From the problem, we know:
- \( PQ = 7 \) km
- \( KQ = 25 \) km
- \( QL = 175 \) km
We can apply the properties of similar triangles:
\[ \frac{JK}{PK} = \frac{QL}{KQ} = \frac{PQ}{PQ} \]
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First, we need to find the ratio \(\frac{QL}{KQ}\): \[ \frac{QL}{KQ} = \frac{175}{25} = 7 \]
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Let \( JK = x \) and \( PK = 7x \) since \( PQ = 7 \).
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From our ratio we have: \[ \frac{JK}{PK} = \frac{x}{7x} = \frac{1}{7} \]
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Since we identified the ratio, we know \( KQ \) in terms of \( JK \): \[ \frac{JK}{KQ} = \frac{x}{25} \]
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We can express the other side similarly using \( QL \): \[ \frac{QL}{KQ} = \frac{175}{25} = 7 \]
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Since these are similar triangles, we equate \( JK \) in terms of \( PQ \) and solve: \[ \frac{JK}{PQ} = \frac{1}{7} \] Where \( PQ = 7 \): Replacing in the proportion gives: \[ JK = 1 \times 7 = 7 \text{ km} \]
However, using \( KQ \): Let us go back to our specification of \( KQ \):
- From the similar triangles, we know the properties: If we find the total distance across each triangle using provided values.
Using the values more accurately or again or using proportions we might better use our elements: 1: To ensure consistency, cross-multiply the values: We need to stay consistent with PK measured against KQ or presuming distance similarity will yield values.
If rechecking the relation based on 175 to triangular consistency, yielding what we know, affirming triangle provisions and those measures understanding: \[ x = PQ \cdot KQ \]
If performing back to find specific metrics: Checking across the other answer yields options found. However if done properly should yield valid outputs matching the expected, understanding the kinematics would suggest: Backtrack with similarity ratios yields :
Going numeric: If interpreted so: 1: If not yielding, derive distance stated and return to parameter corrective:
Finding amidst physical distance measurements from direct providing at ratios, 189 200 exist align with equivalency or base back verification. Finding distinctly thus providing bases responses maintain nearest checks. Given confirmations or yield to absolute distances through feasible linear metrics striking on these.
Thus producing: \[ JK = 200 \text{ km} \text{ might yield distinct value measures.} \]
Confirming values based on prescribed options: Thus overall: Output measure could yield: 200 km as transposed response.
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