Question

Seth is using the figure shown below to prove Pythagorean Theorem using triangle similarity.

In the given triangle ABC, angle A is 90° and segment AD is perpendicular to segment BC.

The figure shows triangle ABC with right angle at A and segment AD. Point D is on side BC.

Part A: Identify a pair of similar triangles. (2 points)

Part B: Explain how you know the triangles from Part A are similar. (4 points)

Part C: If DB = 9 and DC = 4, find the length of segment DA. Show your work. (4 points)

Answer 1

@Ultrilliam
@Hero

Answer 2

When you are given a setup like this where you are given a right triangle where \(\triangle{ABC}\)
\(\overline{AD} \perp \overline{BC}\) this means two smaller right triangles will be created which are similar to each other. To figure out the corresponding angles, you have to know that the smaller angles of angle \(A\) will add to get \(90^{\circ}\) so let the smaller angles of \(\angle A\) be \(x\) and \(y\) we know that since \(\angle A\) is \(90^{\circ}\) then \(x + y = 90^{\circ}\)

Answer 3

Now we also know for the smaller right triangles, the other two angles of a triangle othar than angle \(D\) which we know is a right angle will add to 90 degrees.

Answer 4

So that's how we know those other angles are also \(x\) and \(y\).

Answer 5

And so knowing all this, how can you write the two pairs of similar right triangles? Technically you can write it as three pair of similar right triangles.

Answer 6

The two pear of similar right triangles is ADB and ADC? I'm not sure if this is the correct order.

Answer 7

You right it by matching up the angles which is X, 90, Y

Answer 8

is x + y = 45 + 45

Answer 9

So you correspond the angle name in that order.

Answer 10

Not necessarily. x and y are complementary angles whose sum is 90 degrees. we don't know their values yet.

Answer 11

ok

Answer 12

Care to give it another shot as far as writing the similar triangles?

Answer 13

ADB and CDA?

Answer 14

Correct

Answer 15

How would I find part C?

Answer 16

To find the measure of DA, what you have to do is setup a proportion. Let me research the term for this real fast so you know what this concept is called.

Answer 17

ok

Answer 18

The concept is called Geometric Mean

Answer 19

How would I use it in this problem?

Answer 20

Do i find the geometric mean of 9 and 4

Answer 21

One moment...

Answer 22

To find Geometric Mean you setup the following proportion:
\(\dfrac{BC}{AD} = \dfrac{AD}{DB}\)

Answer 23

Oops Hang on.

Answer 24

\(\color{#0cbb34}{\text{Originally Posted by}}\) @Hero
Created with RaphaëlReply Using Drawing

To find Geometric Mean you setup the following proportion:
\(\dfrac{DC}{AD} = \dfrac{AD}{DB}\)
\(\color{#0cbb34}{\text{End of Quote}}\)

Answer 25

that equation equals: 4/AD = AD/9

Answer 26

Replace AD with \(x\)

Answer 27

oh so 4/x = x/9

Answer 28

Then cross multiply to solve for \(x\)

Answer 29

this equals 6

Answer 30

What equals 6?

Answer 31

x?

Answer 32

Yes but what segment equals 6

Answer 33

DA

Answer 34

or AD

Answer 35

Correct.

Answer 36

thank you

Answer 37

Great job