Question

similar triangles

Answer 1

If triangles are similar, the lengths of corresponding sides are in the same proportion.

Answer 2

can you write the proportions ?

Answer 3

no:/

Answer 4

try to do what you did in the other problems

Pick a side in the red triangle.

Answer 5

From the statement of the similar triangles, can you tell which sides are corresponding from one triangle to the other?

Answer 6

20/16

Answer 7

good, that is shortest red / shortest blue

Answer 8

Yes, and then p/24 right?

Answer 9

any idea what p "matches up with" in the blue triangle ?

Answer 10

yes, the p/24 is good.
now set them equal and solve for p

Answer 11

multiply by 6 too?

Answer 12

first write the proportion
\[ \frac{p}{24}= \frac{20}{16}\]
what do you multiply on the left side so that the 24 in the bottom "goes away"

Answer 13

6

Answer 14

the idea is 24/24 is 1
in other words 24* 1/24 * p is p
so we multiply both sides by 24 (we have to do the same thing to both sides to keep things equal)

Answer 15

30

Answer 16

yes

Answer 17

Thank you for all your help

Answer 18

Since there is an unknown side length, and we are taught in Geometry that we cannot conclude anything about segment lengths or angle measures from figures, unless we are specifically told to do so, looking at the two triangles and stating that the sides with lengths 16 and 20 are the shortest sides is not the correct way to do it. Side p could be shorter. Since its length is unknown, we cannot come to a conclusion about it.

In fact, the sides measuring 16 and 20 were corresponding, but we know it with certainty for a different reason. The statement of similarity of these triangles is:

\(\triangle ABC \sim \triangle XYZ\)

\(\triangle \color{red}{AB}C \sim \triangle \color{red}{XY}Z\) : sides AB and XY are corresponding sides

\(\triangle A\color{red}{BC} \sim \triangle X\color{red}{YZ}\) : sides BC and YZ are corresponding sides

\(\triangle \color{red}{A}B\color{red}{C} \sim \triangle \color{red}{X}Y\color{red}{Z}\) : sides AC and XZ are corresponding sides

We know which sides are corresponding sides by the position of the names of the vertices in the statement of similarity.

Length p is the length of side XY.
Side XY corresponds to side AB, and AB has length 24.

That gives us:

\(\dfrac{p}{24} \)

Now we need two known corresponding lengths to establish the proportion we need.

Side XZ corresponds to side AC.
The lengths of XZ and AC are 20 and 16, respectively.
That gives us the ratio:

\(\dfrac{20}{16} \)

A proportion is setting two ratios equal to each other.
Now we can write the proportion and solve it:

\( \dfrac{p}{24} = \dfrac{20}{16} \)

Reduce the right ratio:

\( \dfrac{p}{24} = \dfrac{5}{4} \)

Multiply both sides by the reciprocal of \( \dfrac{1}{24} \), which is 24:

\(\dfrac{24}{24} p = \dfrac{120}{4}\)

Simplify both sides:

\( p = 30\)