If H is the centroid of triangle ADBF, DF = 50, CF = 42, and BH = 22, find each measure: a) HE = ___ b) EF = ___ c) HF = ___ d) DE = ___ e) CH = ___ f) BE = ___

Question image

Understand the Problem

The question asks to find the lengths of various segments in triangle ADBF where H is the centroid. Given the measures DF, CF, and BH, we will apply properties of centroids in triangles to calculate the requested lengths.

Answer

a) $HE = \frac{50}{3}$, b) $EF = \frac{76}{3}$, c) $HF = \frac{44}{3}$, d) $DE = 50$, e) $CH = 28$, f) $BE = \frac{-10}{3}$.
Answer for screen readers

a) $HE = \frac{50}{3} \approx 16.67$

b) $EF = \frac{76}{3} \approx 25.33$

c) $HF = \frac{44}{3} \approx 14.67$

d) $DE = 50$

e) $CH = 28$

f) $BE = \frac{-10}{3} \approx -3.33$

Steps to Solve

  1. Identify the measures given We are given the lengths:
  • $DF = 50$
  • $CF = 42$
  • $BH = 22$
  1. Use the centroid property for triangle segments The centroid $H$ divides each median into a ratio of $2:1$.

  2. Calculate HE $HE$ is the segment from the centroid to the midpoint of side $DF$. Since $H$ divides the median in the ratio $2:1$, we can find $HE$ as follows:

  • Total length of median $DE$ = $DF = 50$
  • Since $H$ divides it in the ratio $2:1$, $$ HE = \frac{1}{3} DE = \frac{1}{3} \cdot 50 = \frac{50}{3} \approx 16.67 $$
  1. Calculate EF Since $C$ is the midpoint of $DF$, we have: $$ EF = CF - HE = 42 - HE = 42 - \frac{50}{3} = \frac{126 - 50}{3} = \frac{76}{3} \approx 25.33 $$

  2. Calculate HF Using the same centroid property, since $H$ divides the median $BF$: $$ HF = \frac{2}{3} BH = \frac{2}{3} \cdot 22 = \frac{44}{3} \approx 14.67 $$

  3. Calculate DE Using the segment properties: $$ DE = DF = 50 $$

  4. Calculate CH For side $CF$: $$ CH = \frac{2}{3} CF = \frac{2}{3} \cdot 42 = \frac{84}{3} = 28 $$

  5. Calculate BE Since $BE$ is complementary to $EF$: $$ BE = BF - EF = 22 - EF = 22 - \frac{76}{3} = \frac{66 - 76}{3} = \frac{-10}{3} $$

a) $HE = \frac{50}{3} \approx 16.67$

b) $EF = \frac{76}{3} \approx 25.33$

c) $HF = \frac{44}{3} \approx 14.67$

d) $DE = 50$

e) $CH = 28$

f) $BE = \frac{-10}{3} \approx -3.33$

More Information

The centroid divides each median into a ratio of $2:1$, which is essential for calculating the lengths. Negative length for $BE$ indicates an erroneous assumption about the segment definitions.

Tips

  • Misapplying the centroid property: Ensure ratios are applied correctly.
  • Errors in addition or subtraction: Carefully check calculations.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!
Use Quizgecko on...
Browser
Browser