In \( \triangle EFG \), \( K \) is the centroid. If \( GH = 18 \), find \( KH \). ### Diagram Explanation The diagram illustrates a triangle \( \triangle EFG \) with a centroid \( K \). The centroid is connected to the vertices \( E \), \( F \), and \( G \) by lines \( KE \), \( KF \), and \( KG \). Additionally, the line \( GH \) is drawn from vertex \( G \) to a point \( H \) on side \( EF \), intersecting the median \( FK \) at \( K \). The length \( GH \) is given as 18 units. The problem asks to find \( KH \), the segment from \( K \) to \( H \). ### Solution In a triangle, the centroid divides each median into a ratio of 2:1. Therefore, if \( GH = 18 \), then: \[ KH = \frac{1}{3} \times GH = \frac{1}{3} \times 18 = 6 \] So, the length of \( KH \) is 6 units. In triangle \( \triangle GHI \), \( M \) is the intersection of the three medians. If \( JM = 2x + 5 \) and \( MI = 9x - 10 \), find \( x \), \( JM \), and \( MI \). Diagram Explanation: The diagram shows triangle \( \triangle GHI \) with points \( J \), \( K \), and \( L \) as the midpoints of sides \( \overline{HG} \), \( \overline{HI} \), and \( \overline{GI} \), respectively. The medians \( \overline{HJ} \), \( \overline{GK} \), and \( \overline{IL} \) intersect at point \( M \). - \( x = \) [blank box] - \( JM = \) [blank box] - \( MI = \) [blank box]

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

Part A: In triangle EFG, is the centriod. If GH = 18, Find KH.

Part B:

In △GHI,△GHI, MM is the intersection of the three medians. If JM=2x+5JM=2x+5 and MI=9x−10,MI=9x−10, find x,JM,x,JM, and MI.MI.

In \( \triangle EFG \), \( K \) is the centroid. If \( GH = 18 \), find \( KH \).

### Diagram Explanation

The diagram illustrates a triangle \( \triangle EFG \) with a centroid \( K \). The centroid is connected to the vertices \( E \), \( F \), and \( G \) by lines \( KE \), \( KF \), and \( KG \). Additionally, the line \( GH \) is drawn from vertex \( G \) to a point \( H \) on side \( EF \), intersecting the median \( FK \) at \( K \). The length \( GH \) is given as 18 units. The problem asks to find \( KH \), the segment from \( K \) to \( H \).

### Solution

In a triangle, the centroid divides each median into a ratio of 2:1. Therefore, if \( GH = 18 \), then:

\[ KH = \frac{1}{3} \times GH = \frac{1}{3} \times 18 = 6 \]

So, the length of \( KH \) is 6 units.

In triangle \( \triangle GHI \), \( M \) is the intersection of the three medians. If \( JM = 2x + 5 \) and \( MI = 9x - 10 \), find \( x \), \( JM \), and \( MI \).

Diagram Explanation:
The diagram shows triangle \( \triangle GHI \) with points \( J \), \( K \), and \( L \) as the midpoints of sides \( \overline{HG} \), \( \overline{HI} \), and \( \overline{GI} \), respectively. The medians \( \overline{HJ} \), \( \overline{GK} \), and \( \overline{IL} \) intersect at point \( M \).

- \( x = \) [blank box]
- \( JM = \) [blank box]
- \( MI = \) [blank box]

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

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