Answer:
Part 1)
a) True (see the explanation)
b) True (see the explanation)
Part 2) The width of the river is 64.75 units
Part 3) see the explanation
Part 4) see the explanation
Step-by-step explanation:
The question is English
The images are in the order of the questions
1. Reason whether the following statements are true or false. (In either case, write down why)
a. Two equilateral triangles are always similar
b. The value of x is 4 cm
2. Calculate the width of the river
3. In the following figure, indicate which triangles are similar and which are not.
4. How tall is the statue?
Part 1) Reason whether the following statements are true or false. (In either case, write down why)
Part a) Two equilateral triangles are always similar
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent.
An equilateral triangle has three equal sides and three equal interior angles (the measure of each interior angle is 60 degrees)
so
Two equilateral triangles will always have the corresponding congruent angles, therefore they will always be similar
therefore
The statement is true
Part b) The value of x is 4 cm
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent.
In this problem
The smaller triangle is similar with the larger triangle by AA Similarity Theorem
so
[tex]\frac{1.5}{1.5+3}=\frac{2}{x+2}[/tex]
solve for x
[tex]\frac{1}{3}=\frac{2}{x+2}[/tex]
[tex]x+2=6\\x=4\ cm[/tex]
therefore
The statement is true
Part 2) Calculate the width of the river
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent.
In this problem
The two triangles are similar by AA Similarity Theorem
Applying proportion
[tex]\frac{x}{7}=\frac{37}{4}[/tex]
solve for x
[tex]x=\frac{37}{4}(7)[/tex]
[tex]x=64.75\ units[/tex]
Part 3) In the following figure, indicate which triangles are similar and which are not
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent.
In this problem
Triangle ABO is similar with triangle A'B'O by AA Similarity Theorem (side AB is parallel to side A'B')
Triangle ABO is not similar with triangle A''B''O (corresponding angles are not congruent)
Triangle A'B'O is not similar with triangle A''B''O (corresponding angles are not congruent)
Part 4) How tall is the statue?
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent.
In this problem
The smaller triangle and the larger triangle are similar by AA Similarity Theorem
so
Applying proportion
Let
h ----> the height of the statue from the pedestal in meters
[tex]\frac{h}{2.1-1.6}=\frac{4.6+0.9}{0.9}[/tex]
solve for x
[tex]\frac{h}{0.5}=\frac{5.5}{0.9}[/tex]
[tex]h=\frac{5.5}{0.9}(0.5)[/tex]
[tex]h=3.06\ m[/tex]
The height of the statue from the ground is
[tex]3.06+1.6=4.66\ m[/tex]
Round to the nearest tenth
[tex]4.7\ m[/tex]
