GTExl wrote:
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In the diagram DE is parallel to AC. If AC = 10 and DE = 5 and the area of triangle ABC is 40, then what is the area of triangle BDE?.
(A) 8
(B) 10
(C) 12
(D) 14
(E) 20
The two triangles are similar, so the ratio of one side to another will hold for all lengths.
The triangles are similar because they have three identical angles. They share angle B.
Segments AC and DE are parallel (given), cut by two traversal segments AB and BC.
Therefore corresponding angles C and E are congruent, as are corresponding angles A and D. AAA = similar triangles.
Base of large triangle = 10, base of smaller triangle = 5. Ratio of
lengths (not area) is 2:1.
We need height of large triangle to calculate height and area of smaller triangle. Obtain larger triangle's height from larger triangle's given area.
Area = \(\frac{1}{2}b * h\)
40 = \(\frac{1}{2}\)*(10)*(h)
h = 8
Large:small is 2:1 for any one-dimensional length. Height of large triangle is 8; height of small triangle is therefore 4.
Area of smaller triangle BDE is
\(\frac{1}{2}\)*5*4 = 10
Alternatively, take scale factor 2, call it K. For areas, use K\(^2\) (because area = length*length, scale factor is deployed twice, K*K).
Area of 40 = 2\(^2\) * area of smaller triangle
40 = 4 * smaller triangle's area
10 = area of smaller triangle BDE
Answer B
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