7-5 Parts Of Similar Triangles s5c1k

 

7-5 Parts of Similar Triangles Find x.

10. ROADWAYS The intersection of the two roads shown forms two similar triangles. If AC is 382 feet, MP is 248 feet, and the gas station is 50 feet from the intersection, how far from the intersection is the bank?

6.  SOLUTION:   By AA Similarity, the given two triangles are similar. Theorem 7.9 states that if two triangles are similar, the lengths of corresponding angle  bisectors are proportional to the lengths of corresponding sides. Therefore,  SOLUTION:   If two triangles are similar, the lengths of corresponding altitudes are proportional to the lengths of corresponding sides. Let x be the distance between the intersection and the bank.

 

 

8. 

  SOLUTION:   By AA Similarity, the given two triangles are similar. Theorem 7.10 states that if two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides. We know that the segments marked x and 21 are medians because they intersect the opposite side at its midpoint. Therefore, 

Therefore, the distance between the intersection at point B and the bank is about 77 feet. ORGANIZE IDEAS  Find the value of each variable.

 

10. ROADWAYS The intersection of the two roads shown forms two similar triangles. If AC is 382 feet, MP is 248 feet, and the gas station is eSolutions Manual - Powered by Cognero 50 feet from the intersection, how far from the intersection is the bank?

12. 

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SOLUTION:   An angle bisector in a triangle separates the opposite side into two

  Therefore, the distance between the intersection at point B and the bank 7-5 Parts of Similar Triangles is about 77 feet. ORGANIZE IDEAS  Find the value of each variable.

14.  12.  SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

 

 

16. ALGEBRA If  and   are medians,  = 12, WY = 7x – 1, and VX = 4x + 2, find x.

 NQ = 8, PR

SOLUTION:   If two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides.

14.  SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

 

  Substitute.

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  Solve for x.

7-5 Parts of Similar Triangles

16. ALGEBRA If  and   are medians,  = 12, WY = 7x – 1, and VX = 4x + 2, find x.

ALGEBRA Find x.

 NQ = 8, PR

20.  SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

SOLUTION:   If two triangles are similar, the lengths of corresponding medians are proportional to the lengths of corresponding sides.

 

  Substitute.

  Solve for x.

ALGEBRA Find x.

22.  SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

  20.  SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

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24. SPORTS Consider the triangle formed by the path between a batter, center fielder, and right fielder as shown. If the batter gets a hit that

7-5 Parts of Similar Triangles 24. SPORTS Consider the triangle formed by the path between a batter, center fielder, and right fielder as shown. If the batter gets a hit that bisects the triangle at ∠B, is the center fielder or the right fielder closer to the ball? Explain your reasoning. 22.  SOLUTION:   An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

 

SOLUTION:   Right fielder; sample answer: According to the Triangle Bisector Theorem, an angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.  Since the hit bisects the triangle, the sides opposite the batter are 

24. SPORTS Consider the triangle formed by the path between a batter, center fielder, and right fielder as shown. If the batter gets a hit that bisects the triangle at ∠B, is the center fielder or the right fielder closer to the ball? Explain your reasoning.

proportional to the other two sides, such as

or

 

  Substituting,

 Since 

 is slightly greater than 1, CH

is slightly longer than RH. 

  Therefore, the right fielder is closer to the hit. 33. ANALYZE RELATIONSHIPS The perimeter of

 is 94 units. 

 bisects ∠PQR. Find PS and RS. SOLUTION:   Right fielder; sample answer: According to the Triangle Bisector Theorem, an angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.  Since the hit bisects the triangle, the sides opposite the batter are  eSolutions Manual - Powered by Cognero proportional to the other two sides, such as

or

 

SOLUTION:   The perimeter of triangle PQR = 94.  We know that PQ + QR + PR = 94.

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 Since 

Substituting,

 is slightly greater than 1, CH

is slightly longer than RH.    7-5 Parts of Similar Triangles Therefore, the right fielder is closer to the hit.

  Since PS is about 18.4, RS is about 42.4 – 18.4 or 24.

33. ANALYZE RELATIONSHIPS The perimeter of

 is 94 units. 

37. If

, what is the value of x?

 bisects ∠PQR. Find PS and RS.

F 14.4 H 20.5 G 15 J 22.5

SOLUTION:   The perimeter of triangle PQR = 94.  We know that PQ + QR + PR = 94.

SOLUTION:   If , then the corresponding sides are proportional. In order to more easily determine which sides of the similar triangles correspond to each other, redraw them so that they are oriented in the same direction.   

  Substitute given values and solve for PR: 22.4 + 29.2 + PR = 94 51.6 + PR = 94 PR = 42.4

  Form a proportion using similar triangles substitute given values:

and

and

  Now, set up a proportion and solve for x.   

    The correct choice is J.

  Since PS is about 18.4, RS is about 42.4 – 18.4 or 24. 37. If

, what is the value of x?

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14.4 H 20.5 G 15

38. ACT/SAT In  

,

is 1.5 times as long as

          Which of the following statements must be true? 

.

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7-5 Parts of Similar Triangles     The correct choice is J. 38. ACT/SAT In  

,

is 1.5 times as long as

.

          Which of the following statements must be true?  is 1.5 times as long as          I.   is 1.5 times as long as         II.  is 1.5 times as long as         III.    A   I only B    II only C   III only D   II and III only E    I, II, and III

. . .

SOLUTION:   According to the Triangle Angle Bisector Theorem, an angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides. Therefore, the only correct statement is I, as it states that the two sides of the triangle are in the same proportion to the side cut by the angle bisector.  The correct  response is A.

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