in △hjk, hd = 21 and bk = 18. find each length.
5. hb
6. bd
7. ck
8. cb

Question

Accepted Answer

# Explanation:
## Step1: Identify Centroid Property
In a triangle, the centroid (point $ B $ here) divides each median into a ratio of $ 2:1 $, with the longer segment being closer to the vertex. A median connects a vertex to the midpoint of the opposite side.
## Step2: Solve for $ HB $ (Question 5)
For median $ HD $ (from $ H $ to midpoint $ D $ of $ JK $), centroid $ B $ divides $ HD $ such that $ HB:BD = 2:1 $. Let $ HB = 2x $ and $ BD = x $. Then $ HD = HB + BD = 2x + x = 3x $. Given $ HD = 21 $, so $ 3x = 21 \Rightarrow x = 7 $. Thus, $ HB = 2x = 14 $.
## Step3: Solve for $ BD $ (Question 6)
From Step 2, $ BD = x = 7 $ (or since $ HD = 21 $ and $ HB = 14 $, $ BD = HD - HB = 21 - 14 = 7 $).
## Step4: Solve for $ CK $ (Question 7)
For median $ JK $ (wait, no, $ BK $ is part of median from $ K $ to midpoint $ C $ of $ HJ $? Wait, the marks: $ HC = CJ $ (since $ HJ $ has two marks), $ JE = EK $ (two marks on $ JK $? Wait, no, $ HE = EK $ (three marks on $ HE $ and two on $ EK $? Wait, no, the triangle: $ HJ $ has two marks (so $ C $ is midpoint, $ HC = CJ $), $ JK $ has two marks (so $ D $ is midpoint, $ JD = DK $), $ HK $ has three marks on $ HE $ and two on $ EK $? Wait, no, the centroid is where medians intersect. So $ BK $ is part of the median from $ K $ to midpoint $ C $ of $ HJ $. Wait, no, $ BK $ is from $ B $ to $ K $, but the median from $ K $ would go to midpoint $ C $ of $ HJ $. Wait, actually, in centroid, each median is divided by centroid into $ 2:1 $. So if $ BK $ is the segment from centroid $ B $ to vertex $ K $, then the median from $ K $ to midpoint $ C $ of $ HJ $ has $ BK:BC = 2:1 $. Given $ BK = 18 $, let $ BC = x $, then $ BK = 2x \Rightarrow 2x = 18 \Rightarrow x = 9 $, so the median length is $ BK + BC = 18 + 9 = 27 $, but we need $ CK $? Wait, no, $ CK $ is the length from $ C $ to $ K $, which is the median. Wait, no, $ C $ is midpoint of $ HJ $, so the median from $ K $ to $ C $ is $ KC $, and centroid $ B $ divides $ KC $ into $ KB:BC = 2:1 $. So $ KC = KB + BC = 18 + 9 = 27 $? No, wait, no: centroid divides the median into $ 2:1 $, with the longer part (2 parts) being from vertex to centroid. So median from $ K $ to $ C $ (midpoint of $ HJ $): $ BK $ is the part from $ B $ (centroid) to $ K $ (vertex), so $ BK $ is $ \frac{2}{3} $ of the median $ KC $. Wait, no: centroid is $ \frac{2}{3} $ from vertex, $ \frac{1}{3} $ from midpoint. So if $ BK $ is from centroid $ B $ to vertex $ K $, then $ BK = \frac{2}{3} \times $ median length. Wait, I think I messed up earlier. Let's correct: In a median, the distance from vertex to centroid is $ \frac{2}{3} $ of the median, and from centroid to midpoint is $ \frac{1}{3} $. So for median from $ K $ to midpoint $ C $ of $ HJ $: $ BK = \frac{2}{3} \times KC $, and $ BC = \frac{1}{3} \times KC $. Given $ BK = 18 $, then $ \frac{2}{3} KC = 18 \Rightarrow KC = 18 \times \frac{3}{2} = 27 $. But $ CK $ is the same as $ KC $, so $ CK = 27 $? Wait, no, $ CK $ is from $ C $ to $ K $, which is the median, so yes, $ CK = 27 $. Wait, but maybe I had it reversed. Let's re-express: Let median be $ KC $ (from $ K $ to $ C $, midpoint of $ HJ $). Centroid $ B $ is on $ KC $, so $ KB:BC = 2:1 $. So $ KB = 2 $ parts, $ BC = 1 $ part. Given $ KB = 18 $, so 2 parts = 18, 1 part = 9. Thus, $ KC = KB + BC = 18 + 9 = 27 $, so $ CK = KC = 27 $.
## Step5: Solve for $ CB $ (Question 8)
From Step 4, since $ BK:BC = 2:1 $ and $ BK = 18 $, then $ BC = \frac{1}{2} BK = \frac{1}{2} \times 18 = 9 $.

# Answers:
5. $ \boldsymbol{14} $
6. $ \boldsymbol{7} $
7. $ \boldsymbol{27} $
8. $ \boldsymbol{9} $

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QUESTION IMAGE

Question

Explanation:

Step1: Identify Centroid Property

Step2: Solve for \( HB \) (Question 5)

Step3: Solve for \( BD \) (Question 6)

Step4: Solve for \( CK \) (Question 7)

Step5: Solve for…

Answer: