Questions: Exercises. In triangle ABC, AU=16, BU=12, and CF=18. Find each measure. 1. UD 2. EU 3. CU 4. AD 5. UF 6. BE

Transcript text: Exercises. In $\triangle A B C, A U=16, B U=12$, and $C F=18$. Find each measure. 1. UD 2. $E U$ 3. $C U$ 4. $A D$ 5. $U F$ 6. $B E$

Solution

Solution Steps

Step 1: Find UD

Since CE = CA and CD = CB, it follows that E and D are midpoints of the sides CA and CB respectively. Therefore, ED is a midsegment of triangle ABC. A midsegment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side. Since ED is parallel to AB, and U is the centroid, UD = (1/3) * BU and UD = (1/2) * ED = (1/2) * (1/2) * AB. Therefore, UD = (1/3) * 12 = 4.

Step 2: Find EU

Since AF = AB and AE = AC, it follows that F and E are midpoints of the sides AB and AC respectively. Therefore, FE is a midsegment of triangle ABC. Since FE is parallel to BC and U is the centroid, we have EU = (1/3) * CU and EU = (1/2) * FE = (1/2) * (1/2) * BC. EU = (1/3) * AU. Therefore, EU = (1/3) * 16 = 16/3.

Step 3: Find CU

Since CF is a median, and U is the centroid, CU = (2/3) * CF. Therefore, CU = (2/3) * 18 = 12.