Question

Use the properties of a centroid and the figure for examples 4-7. BD, CF, and AE are medians of △ABC. BD = 21 and GE = 5. Find each length.

A. GE = 3.75, GD = 17.25
B. GE = 4.5, GD = 16.5
C. GE = 6.75, GD = 14.25
D. GE = 7.5, GD = 13.5

Answer 1

The centroid of a triangle divides medians into a 2:1 ratio. GD is twice the length of GE, with GD + GE equaling the total median length. There seems to be a mistake in the choices provided as none match GD = 16 and GE = 5.

Step-by-step explanation:

The student has asked a question related to the properties of a centroid. In a triangle, the centroid is the point where the three medians intersect, and it divides each median into segments with a 2:1 ratio, with the longer segment being closest to the vertex of the triangle. Given that BD is a median of △ABC and BD = 21, and GE = 5, we can determine that GD is the longer segment from the centroid to the vertex, which would be twice the length of GE.

To find the length of GD, we use the ratio 2:1. Since GE is one part, GD must be two parts, making GD 2*5 = 10. To find the complete median length, we add GD and GE together, so the complete median length GD + GE is 10 + 5 = 15. But we know that BD (the complete median length) is 21, so GD must be 21 - 5, which gives us GD = 16. Now, the correct answer is the one where GE is 5 and GD is 16, which is option B. However, in the given options, there's no such pairing. It seems there is a mistake in the provided choices.