Let abc be a triangle with centroid g. points l, m, and n are the midpoints of sides bc, ca, and ab respectively. let d be the foot of the altitude from a to bc and let k be the foot of the altitude from l to mn.

(a) show that ad/lk=2

(b) show that triangleadg is similar to trianglelkg

(c) show that d, g and k are collinear and that dg/gk=2

See explanation

Step-by-step explanation:

1. MN is the midline of the triangle ABC. By the midline theorem, MN is parallel to BC. AD is perpendicular to BC, KL is perpendicular to MN, so KL is perpendicular to BC. Two perpendicular lines to BC are parallel, thus

AD ║ KL.

In quadrilateral EDLK,

ED ║ KL

DL ║ EK,

so EDLK is a parallelogram.

Opposite sides of the parallelogram are congruent, so

Midline MN of the triangle ABC divides the height AD into two congruent parts AE and ED. So,

2. Since AD ║ KL, then

∠DAG ≅ ∠GLK

as alternate interior angles when parallel lines AD and KL are cut by transversal AL.

Since G is centroid, then

By SAS similarity theorem,

3. If   then

∠DGA ≅ ∠KGL

Since points A, G and L lie on the same line AL, then points D, G and K lie on the same line DK, so they are collinear

x = 10.1

step-by-step explanation:

all of the angles of a tirangle add up to 180º

180 - 65 = 115 (since we already know the value of one angle)

(2x + 7) + (8x + 7) = 115

(combins like terms) 10x + 14 = 115

(isolate the variable subtract 14 from both sides): 10x = 101

(find the variable divide both sides by 10): x = 101/10

x = 10.1