Question

1 two right triangles are similar, with side lengths of one triangle being 5, 12, and 13. if the shortest side of the second triangle is 10, what is the length of the longest of the second tri a 26 b 40 c 20 d 30 2 the length of a buildings shadow is 20 meters at the same time that a 3 - meter stick casts a shadow of 4 meters. using similar triangles, what is the height of the building? a 15 meters b 10 meters c 5 meters d 20 meters 3 what does the aa criterion stand for in triangle similarity? a all - side - angle b axis - angle c angle - area d angle - angle 4 a tree casts a shadow of 12 feet, while a 4 - foot stick casts a shadow of 6 feet. what is the height of the tree? a 12 feet b 6 feet c 8 feet d 10 feet 5 what is the criterion that determines triangle similarity if two pairs of angles are congruent? a aa - criterion b sss criterion c sas criterion d right triangle criterion 6 in \\( \triangle ghi, gh = 12, hi = 16 \\) and \\( \angle c = 60 ^ { \circ } \\) in \\( \triangle jkl, jk = 4, kl = 5 \\) and \\( \angle j = 60 ^ { \circ } \\) are these triangles similar, and by which criterion? a no, the sides are not proportional b no, the angles are not equal c yes, by the aa criterion d yes, by the sas criterion 7 in \\( \triangle abc \sim \triangle def, ab = 10, ac = 15, \\) and \\( de = 6 \\) what is the length of ef? a 10 a 25

Explanation:

1. Problem 1:

• Explanation:
• Step1: Find the scale factor
• The shortest side of the first triangle is \(5\), and the shortest side of the second triangle is \(10\). The scale factor \(k=\frac{10}{5} = 2\).
• Step2: Find the longest side of the second triangle
• The longest side of the first triangle is \(13\). Using the property of similar triangles (corresponding sides are proportional), if the scale factor is \(k = 2\), then the longest side of the second triangle \(L=13\times k\). Substituting \(k = 2\), we get \(L = 13\times2=26\).
• Answer: a. 26

1. Problem 2:

• Explanation:
• Step1: Set up the proportion
• Let the height of the building be \(h\). For similar triangles, \(\frac{h}{20}=\frac{3}{4}\) (height of object to length of shadow).
• Step2: Solve for \(h\)
• Cross - multiply: \(4h=3\times20\). Then \(h=\frac{3\times20}{4}=15\).
• Answer: a. 15 meters

1. Problem 3:

• Brief Explanations:
• The AA (Angle - Angle) criterion for triangle similarity states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
• Answer: d. Angle - Angle

1. Problem 4:

• Explanation:
• Step1: Set up the proportion
• Let the height of the tree be \(h\). Using the property of similar triangles (height to shadow length ratio), \(\frac{h}{12}=\frac{4}{6}\).
• Step2: Solve for \(h\)
• Cross - multiply: \(6h = 4\times12\). Then \(h=\frac{4\times12}{6}=8\).
• Answer: c. 8 feet

1. Problem 5:

• Brief Explanations:
• The AA (Angle - Angle) criterion determines triangle similarity when two pairs of angles are congruent.
• Answer: a. AA Criterion

1. Problem 6:

• Brief Explanations:
• We are not given enough information about the angles or side - angle - side relationships to confirm similarity. Just having some side lengths and one angle is not sufficient as the side lengths may not be in proportion and the angles may not be corresponding in a way that satisfies similarity criteria.
• Answer: b. No, the angles are not equal

1. Problem 7:

• Explanation:
• Step1: Assume \(\triangle ABC\sim\triangle DEF\) (assuming the correct similarity notation). For similar triangles \(\frac{AB}{DE}=\frac{AC}{DF}\)
• Let \(DF=x\). Given \(AB = 10\), \(AC = 15\), \(DE = 6\). Then \(\frac{10}{6}=\frac{15}{x}\).
• Step2: Solve for \(x\)
• Cross - multiply: \(10x=6\times15\). So \(x=\frac{6\times15}{10}=9\).
• Answer: 9 (assuming the correct similarity setup as the problem statement has some notation issues, but based on the proportion of sides for similar triangles)

Answer:

1. Problem 1:

• Explanation:
• Step1: Find the scale factor
• The shortest side of the first triangle is \(5\), and the shortest side of the second triangle is \(10\). The scale factor \(k=\frac{10}{5} = 2\).
• Step2: Find the longest side of the second triangle
• The longest side of the first triangle is \(13\). Using the property of similar triangles (corresponding sides are proportional), if the scale factor is \(k = 2\), then the longest side of the second triangle \(L=13\times k\). Substituting \(k = 2\), we get \(L = 13\times2=26\).
• Answer: a. 26

1. Problem 2:

• Explanation:
• Step1: Set up the proportion
• Let the height of the building be \(h\). For similar triangles, \(\frac{h}{20}=\frac{3}{4}\) (height of object to length of shadow).
• Step2: Solve for \(h\)
• Cross - multiply: \(4h=3\times20\). Then \(h=\frac{3\times20}{4}=15\).
• Answer: a. 15 meters

1. Problem 3:

• Brief Explanations:
• The AA (Angle - Angle) criterion for triangle similarity states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
• Answer: d. Angle - Angle

1. Problem 4:

• Explanation:
• Step1: Set up the proportion
• Let the height of the tree be \(h\). Using the property of similar triangles (height to shadow length ratio), \(\frac{h}{12}=\frac{4}{6}\).
• Step2: Solve for \(h\)
• Cross - multiply: \(6h = 4\times12\). Then \(h=\frac{4\times12}{6}=8\).
• Answer: c. 8 feet

1. Problem 5:

• Brief Explanations:
• The AA (Angle - Angle) criterion determines triangle similarity when two pairs of angles are congruent.
• Answer: a. AA Criterion

1. Problem 6:

• Brief Explanations:
• We are not given enough information about the angles or side - angle - side relationships to confirm similarity. Just having some side lengths and one angle is not sufficient as the side lengths may not be in proportion and the angles may not be corresponding in a way that satisfies similarity criteria.
• Answer: b. No, the angles are not equal

1. Problem 7:

• Explanation:
• Step1: Assume \(\triangle ABC\sim\triangle DEF\) (assuming the correct similarity notation). For similar triangles \(\frac{AB}{DE}=\frac{AC}{DF}\)
• Let \(DF=x\). Given \(AB = 10\), \(AC = 15\), \(DE = 6\). Then \(\frac{10}{6}=\frac{15}{x}\).
• Step2: Solve for \(x\)
• Cross - multiply: \(10x=6\times15\). So \(x=\frac{6\times15}{10}=9\).
• Answer: 9 (assuming the correct similarity setup as the problem statement has some notation issues, but based on the proportion of sides for similar triangles)