Question

1. state if the triangles in each pair are similar. if so, state how they are similar.

a. sss similarity
b. aa similarity
c. not similar
d. sas similarity

1. jack wants to find the height of an olive tree for a science project. jack is two meters tall. he stands six meters away from the tree, and his shadow is four meters long. which proportion is correct?

a. $\frac{4}{6}=\frac{2}{x}$
b. $\frac{4}{10}=\frac{2}{x}$
c. $\frac{4}{2}=\frac{2}{x}$
d. $\frac{x}{6}=\frac{2}{4}$

Explanation:

Question 14

Step1: Calculate side - length ratios

For \(\triangle DCB\) and \(\triangle QPB\), we have \(\frac{DB}{PB}=\frac{40}{24}=\frac{5}{3}\), \(\frac{CB}{QB}=\frac{30}{18}=\frac{5}{3}\). Also, \(\angle DBC=\angle QBP\) (vertically - opposite angles).

Step2: Apply similarity criterion

Since two sides of one triangle are proportional to two sides of the other triangle and the included angles are equal, by the SAS (Side - Angle - Side) similarity criterion, the two triangles are similar.

Step1: Identify similar triangles

The triangle formed by the tree, its shadow, and the line from the top of the tree to the end of the shadow is similar to the triangle formed by Jack, his shadow, and the line from the top of his head to the end of his shadow.

Step2: Set up the proportion

The ratio of the height to the base of the two similar triangles is equal. The height of Jack is 2m, the length of his shadow is 4m, the distance from Jack to the tree is 6m, and the height of the tree is \(x\). The base of the larger triangle (tree's shadow) is \(6 + 4=10\)m. So the proportion is \(\frac{4}{10}=\frac{2}{x}\).

Answer:

d. SAS similarity

Question 15