Question

Las imágenes están en el orden de las preguntas

Answer 1

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

$\frac{1.5}{1.5+3}=\frac{2}{x+2}$

solve for x

$\frac{1}{3}=\frac{2}{x+2}$

$x+2=6\\x=4\ cm$

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

$\frac{x}{7}=\frac{37}{4}$

solve for x

$x=\frac{37}{4}(7)$

$x=64.75\ units$

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

$\frac{h}{2.1-1.6}=\frac{4.6+0.9}{0.9}$

solve for x

$\frac{h}{0.5}=\frac{5.5}{0.9}$

$h=\frac{5.5}{0.9}(0.5)$

$h=3.06\ m$

The height of the statue from the ground is

$3.06+1.6=4.66\ m$

Round to the nearest tenth

$4.7\ m$