Question

Abdul is making a map of his neighborhood. He knows the following information:

• His home, the middle school, and high school are all on the same street.
• His home, the elementary school, and his friend's house are on the same street.
• The distance between home and the middle school and between home and the elementary school is 3 miles.
• The distance between the high school and the middle school and between his friend's house and the elementary school is 6 miles.
• The angle between the elementary school, middle school, and his home is congruent to the angle between his friend's house, the high school, and his home.

What theorem can Abdul use to determine the two triangles are similar? (6 points)

	Pieces of Right Triangles Similarity Theorem
	Side-Side-Side Similarity Theorem
	Midsegment Theorem
	Side-Angle-Side Similarity Theorem

Answer 1

Solution-

Let the positions of Abdul's home, the middle school, and high school (all of these are on the same street) be denoted by A, B and C respectively.

Let the positions of his home, the elementary school, and his friend's house (are on the same street) be denoted by A, D and E respectively.

Since the distance between home and the middle school and between home and the elementary school is 3 miles.

So, AB = AD = 3 mi ......(1)

Since the distance between the high school and the middle school and between his friend's house and the elementary school is 6 miles.

So, BC = DE = 6 mi .....(2)

Since the angle between the elementary school, middle school, and his home is congruent to the angle between his friend's house, the high school, and his home.

So, <DBA = <ECA.

The figure is shown below-

Since AB = AD = 3 mi and BC = DE = 6mi

So, AC = AB + BC = 3 + 6 = 9mi .....(3)

and AE = AD + DE = 3 +6 ÷ 9mi ......(4)

Now, in ∆ACE and ∆ABD

AB/AC = 3/9 = 1/3 (using equations 1 and 3)

AD/AE = 3/9 = 1/3 (using equations 1 and 4)

<A = common.

So, By Side , Angle and Side ( SAS ) criteria ,

∆ACE and ∆ABD are similar.

Hence, Abdul can use SAS theoram to determine the two triangles are similar.