Chapter 7.6, Problem 1WE

a.

To determine

To find: given proportion, $\frac{r}{s}=\frac{a}{b}$ is correct or not.

Answer to Problem 1WE

No, given proportion is not correct.

Explanation of Solution

Given:

  

  $\frac{r}{s}=\frac{a}{b}$

Concept used:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in same proportion.

If two triangles are similar then the corresponding sides are in same proportion.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Name the vertices of triangles as A , B and C and the parallel line as D, E.

  

In $$ \angle B$△ABC$ and $△DBE$

is common and $DE\left|\right|AC$ therefore DE divides two sides of triangle other than AC proportionally.

Therefore, $\frac{a}{j}=\frac{n}{k}$ .

And so $△ABC\sim △DBE$

Now if the two triangles are similar then the corresponding sides are in same proportion.

Therefore, $\frac{r}{s}=\frac{a}{j}$ .

Hence, $\frac{r}{s}=\frac{a}{b}$ is wrong.

b.

To determine

To find: given proportion, $\frac{j}{a}=\frac{s}{r}$ is correct or not.

Answer to Problem 1WE

yes, given proportion is correct.

Explanation of Solution

Given:

  

  $\frac{j}{a}=\frac{s}{r}$

Concept used:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in same proportion.

If two triangles are similar then the corresponding sides are in same proportion.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Name the vertices of triangles as A , B and C and the parallel line as D, E.

  

In $△ABC$ and $△DBE$

  $\angle B$ is common and $DE\left|\right|AC$ therefore DE divides two sides of triangle other than AC proportionally.

Therefore, $\frac{a}{j}=\frac{n}{k}$ .

And so $△ABC\sim △DBE$

Now if the two triangles are similar then the corresponding sides are in same proportion.

Therefore, $\frac{r}{s}=\frac{a}{j}\Rightarrow \frac{j}{a}=\frac{s}{r}$ .

Hence, $\frac{j}{a}=\frac{s}{r}$ is correct.

c.

To determine

To find: given proportion, $\frac{a}{b}=\frac{n}{t}$ is correct or not.

Answer to Problem 1WE

Yes, given proportion is correct.

Explanation of Solution

Given:

  

  $\frac{a}{b}=\frac{n}{t}$

Concept used:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in same proportion.

If two triangles are similar then the corresponding sides are in same proportion.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Name the vertices of triangles as A , B and C and the parallel line as D, E.

  

  $DE\left|\right|AC$ therefore DE divides two sides of triangle other than AC proportionally.

Therefore, $\frac{a}{b}=\frac{n}{t}$ .

Hence, $\frac{a}{b}=\frac{n}{t}$ is correct.

d.

To determine

To find: given proportion, $\frac{t}{k}=\frac{a}{j}$ is correct or not.

Answer to Problem 1WE

No, given proportion is not correct.

Explanation of Solution

Given:

  

  $\frac{t}{k}=\frac{a}{j}$

Concept used:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in same proportion.

If two triangles are similar then the corresponding sides are in same proportion.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Name the vertices of triangles as A , B and C and the parallel line as D, E.

  

In $△ABC$ and $△DBE$

  $\angle B$ is common and $DE\left|\right|AC$ therefore DE divides two sides of triangle other than AC proportionally.

Therefore, $\frac{a}{b}=\frac{n}{t}$ .

And so $△ABC\sim △DBE$

Now if the two triangles are similar then the corresponding sides are in same proportion.

Therefore, $\frac{a}{j}=\frac{n}{k}$ .

Hence, $\frac{t}{k}=\frac{a}{j}$ is not correct.

e.

To determine

To find: given proportion, $\frac{r}{s}=\frac{n}{k}$ is correct or not.

Answer to Problem 1WE

Yes, given proportion is correct.

Explanation of Solution

Given:

  

  $\frac{r}{s}=\frac{n}{k}$

Concept used:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in same proportion.

If two triangles are similar then the corresponding sides are in same proportion.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Name the vertices of triangles as A , B and C and the parallel line as D, E.

  

In $△ABC$ and $△DBE$

  $\angle B$ is common and $DE\left|\right|AC$ therefore DE divides two sides of triangle other than AC proportionally.

Therefore, $\frac{a}{j}=\frac{n}{k}$ .

And so $△ABC\sim △DBE$

Now if the two triangles are similar then the corresponding sides are in same proportion.

Therefore, $\frac{r}{s}=\frac{n}{k}$ .

Hence, $\frac{r}{s}=\frac{n}{k}$ is correct.

f.

To determine

To find: given proportion, $\frac{b}{j}=\frac{t}{k}$ is correct or not.

Answer to Problem 1WE

Yes, given proportion is correct.

Explanation of Solution

Given:

  

  $\frac{b}{j}=\frac{t}{k}$

Concept used:

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in same proportion.

If two triangles are similar then the corresponding sides are in same proportion.

Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

Name the vertices of triangles as A , B and C and the parallel line as D, E.

  

In $△ABC$ and $△DBE$

  $\angle B$ is common and $DE\left|\right|AC$ therefore DE divides two sides of triangle other than AC proportionally.

Therefore, $\frac{a}{b}=\frac{n}{t}$ .

And so $△ABC\sim △DBE$

Now if the two triangles are similar then the corresponding sides are in same proportion.

Therefore, $\frac{a}{j}=\frac{n}{k}$ .

We have

  $\frac{a}{b}=\frac{n}{t}\Rightarrow \frac{a}{n}=\frac{b}{t}$

And $\frac{a}{j}=\frac{n}{k}\Rightarrow \frac{a}{n}=\frac{j}{k}$

Which gives,

  $\frac{b}{t}=\frac{j}{k}\Rightarrow \frac{b}{j}=\frac{t}{k}$

Hence, $\frac{b}{j}=\frac{t}{k}$ is correct.