Aim 7: What is the right-triangle altitude theorem



Aim 7 & 8: What is the right-triangle altitude theorem? How do we apply it?

Do Now: Construct altitude [pic].

If [pic][pic] [pic], what kind of triangle is WXY?

What kind of angle is [pic]?

How many pairs of similar ∆’s are there?

Name them.

_____________________________________________ONE WAY______________________________________________________________

Given: [pic] is a right [pic], with right [pic]. & altitude [pic]. Complete each proportion.

Theorem I: The altitude to the hypotenuse of a right [pic] is the mean proportional between the segments of the hypotenuse.

So we write [pic] which is [pic]

Theorem II: Each leg of a right [pic] is mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

(Projection is almost like a shadow that gets cast when a flashlight shines on a leg.)

The “big” triangle has two legs- leg “a” & leg “b”, so [pic]

Compares Big ∆ & little ∆ with leg “a” [pic]

Compares Big ∆ & little ∆ with leg “b” [pic]

___________________________________________ANOTHER WAY___________________________________________________________

How can we use the three little bears & BLT’s to write proportions from an altitude drawn in a right triangle? (A different method!)

Given: [pic] is a right [pic], with right [pic] & altitude [pic]. Complete each proportion.

Theorem I: The altitude to the hypotenuse of a right [pic] is the mean proportional between the segments of the hypotenuse.

This is the comparison of Baby Bear and Mama Bear. (Legs & Back)

So we write [pic] which is [pic]

Theorem II: Each leg of a right [pic] is mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

This compares Papa & Baby (Tummy & Legs) This compares Papa and Mama (Tummy & Back).

[pic] or [pic] [pic] or [pic]

Writing Exercise: Which way do you prefer? Explain why.

1. 2. 3. 4.

For questions 1 – 4, use the diagrams above and find the missing segments.

1 . Given: ∆PQR with right [pic] and altitude [pic]. a) [pic] b) [pic] c) [pic]

d) If PS=16, SR=4. QS= ___ e) If SR=4, PR=9, QR= ___

_____________________________________________________________________________________________2. Given: ∆WXY with right [pic] and altitude [pic]. a) [pic] b) [pic] c) [pic] *d) If AW=6, XY=4, XA= __

{Hint: Let XA =m & represent XW in terms of m}

____________________________________________________________________________________________

3. Given: ∆DEF with right [pic] and altitude [pic]. a) [pic] b) [pic] c) [pic] d) If GE=5, GF=4, GD= ___ * e) If FE=9, GD=24, FD= ___

{Hint: Let FD = x & represent FG in terms of x}

____________________________________________________________________________________________

4. Given: ∆KLM with right [pic] and altitude [pic]. If KN=2, NL=8 (simplest radical form for part c &d)

a) KL =____ b) MN=____ c) KM=____ d) LM=____

____________________________________________________________________________________________

5. Prove each theorem when the altitude is drawn to the hypotenuse of a right ∆:

a) the 2 ∆’s formed are similar to the given ∆.

b) each leg of a right ∆ is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

c) the length of the altitude is the mean proportional between the segments of the hypotenuse.

Statements Reasons

1. ∆ ABC is a right ∆ , right [pic]ABC 1. given

2. [pic] [pic] [pic] 2. given

3. [pic]BDA & [pic]BDC are right [pic]’s 3.

4. [pic]ABC [pic] [pic]BDA 4.

5. [pic]A [pic] [pic]A 5.

6. ∆ BAD ~ ∆ ____ 6. a.a. [pic] a.a.

7. [pic] 7.

8. [pic]ABC [pic] [pic]BDC 8.

9. [pic]C [pic] [pic]C 9.

10. ∆ CAB ~ ∆____ 10. a.a. [pic] a.a.

11. [pic] 11. Corresponding sides of ~ ∆’s are in proportion.

12. ∆ BAD ~ ∆ CBD 12. ________________ Property (6, 10)

13. [pic] 13. Corresponding sides of ~ ∆’s are in proportion.

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