2. If P is any point on side AB of triangle ABC and F, K, E are the midpoints of AP, PB, and PC, prove that AFKE ~ AABC. "Exercise 4.14 #10: If the hypotenuse and a leg of one right triangle are proportional to the hypotenuse and a leg of another, then the two triangles are similar. A4 c' a' T' c' - Given: b b' - Prove: AT ~ AT' Statements Reasons Statements Reasons 1. (see above) 1. Given a2 is b2 5. Subtraction a'2 %| b'2 Transformation 2. If 4 quantities are in proportion, then like powers are in proportion. c2 c'2 a2 b2 6. Alternation %3D b2 b'2 b'2 Transformation 3. c2 = a2 + b2; c'2 = a'2 + b'2 7. If 4 quantitie proportion, the are in proportic b. 3. Pythagorean Theorem b' a' a?+b2 4. b2 a'2+b'2 %3D b'2 4. Substitution (step 3 -> 2) 9. AT ~ AT' 9. 1.1. 6. 7. 2.

Elementary Geometry For College Students, 7e
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Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter6: Circles
Section6.CT: Test
Problem 1CT: a If mAB=88, then mACB=____________. b If mAB=92 and C is the midpoint of major are ACB, then...
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Please answer in a two-column proof-statements and reasons.

It is about similar triangles. The question is in the picture. **The question is completely clear, so please don't reject the question.** I have an example of how it should look like in the second picture. Also, it should have ONLY one given.

*If you could* (maybe), please choose some of these statements and use some of these reasons to prove the statements true. (Those below are the ideas for you to use if you want to.)

Definition of ~ triangles

Definition of median

Definition of midpoint

Multiplication

Division

If 4 quantities are in proportion, then like powers are in proportion.

Subtraction Transformation

Alternation Transformation

Pythagorean Theorem

If 2 angles have their sides perpendicular, right side to right side and left side to left side, the angles are equal.

In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

Angles inscribed in the same segment or equal segments are equal.

If a line is drawn parallel to the base of a triangle, it cuts off a triangle similar to the given triangle.

Two isosceles triangles are similar if any angle of one equals the corresponding angle of the other.

C.A.S.T.E. - corresponding angles of similar triangles are equal

C.S.S.T.P. - corresponding sides of similar triangles are proportional

Theorem 57- If two triangles have the three angles of one equal respectively to the three angles of the other, then the triangles are similar

Corollary 57-1 If two angles of one triangle are equal respectively to two angles of another, then the triangles are similar. (a.a.)

Corollary 57-2 Two right triangles are similar if an acute angle of one is equal to an acute angle of the other.

Theorem 58-If two triangles have two pairs of sides proportional and the included angles equal respectively, then the two triangles are similar. (s.a.s.)

Corollary 58-1 If the legs of one right triangle are proportional to the legs of another, the triangles are similar. (l.l.)

Theorem 59- If two triangles have their sides respectively proportional, then the triangles are similar. (s.s.s.)

2. If P is any point on side AB of triangle
ABC and F, K, E are the midpoints of AP,
PB, and PC, prove that AFKE ~ AABC.
Transcribed Image Text:2. If P is any point on side AB of triangle ABC and F, K, E are the midpoints of AP, PB, and PC, prove that AFKE ~ AABC.
"Exercise 4.14 #10: If the hypotenuse and a
leg of one right triangle are proportional to
the hypotenuse and a leg of another, then
the two triangles are similar.
A4
c'
a'
T'
c'
- Given:
b
b'
- Prove: AT ~ AT'
Statements
Reasons
Statements
Reasons
1. (see above)
1. Given
a2
is
b2
5. Subtraction
a'2
%|
b'2
Transformation
2. If 4 quantities are in
proportion, then like
powers are in proportion.
c2
c'2
a2
b2
6. Alternation
%3D
b2
b'2
b'2
Transformation
3. c2 = a2 + b2;
c'2 = a'2 + b'2
7. If 4 quantitie
proportion, the
are in proportic
b.
3. Pythagorean Theorem
b'
a'
a?+b2
4.
b2
a'2+b'2
%3D
b'2
4. Substitution (step 3 -> 2) 9. AT ~ AT'
9. 1.1.
6.
7.
2.
Transcribed Image Text:"Exercise 4.14 #10: If the hypotenuse and a leg of one right triangle are proportional to the hypotenuse and a leg of another, then the two triangles are similar. A4 c' a' T' c' - Given: b b' - Prove: AT ~ AT' Statements Reasons Statements Reasons 1. (see above) 1. Given a2 is b2 5. Subtraction a'2 %| b'2 Transformation 2. If 4 quantities are in proportion, then like powers are in proportion. c2 c'2 a2 b2 6. Alternation %3D b2 b'2 b'2 Transformation 3. c2 = a2 + b2; c'2 = a'2 + b'2 7. If 4 quantitie proportion, the are in proportic b. 3. Pythagorean Theorem b' a' a?+b2 4. b2 a'2+b'2 %3D b'2 4. Substitution (step 3 -> 2) 9. AT ~ AT' 9. 1.1. 6. 7. 2.
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