Right triangle ABC is represented with right angle at vertex B, towards left. Point D lies on side AC. Segment BD is drawn. Angle BDC is marked as right angle.

The figure shows ∆ABC with m∠ABC = 90°.

The first 10 steps to prove that AB2 + BC2 = AC2 are given in the table. Match the remaining steps to their correct sequence in the proof.

Statement Reason
1. Draw
. construction
2. ∠ABC ≅ ∠BDC Angles with the same measure are congruent.
4. ∠BCA ≅ ∠DCB Reflexive Property of Congruence
4.
AA criterion for similarity
5.
Corresponding sides of similar triangles are proportional.
6. BC2 = AC × DC cross multiplication
7. ∠ABC ≅ ∠ADB Angles with the same measure are congruent.
8. ∠BAC ≅ ∠DAB Reflexive Property of Congruence
9.
AA criterion for similarity
10.
Corresponding sides of similar triangles are proportional.
11.
12.
13.
14.
15.
AB2 + BC2 = AC2
Reason: multiplication
AB2 + BC2 = AC × AC
Reason: segment addition
AB2 = AC × AD
Reason: cross multiplication
AB2 + BC2 = AC(AD + DC)
Reason: Distributive Property
AB2 + BC2 = AC × AD + AC × DC
Reason: addition
11
arrowBoth
12
arrowBoth
13
arrowBoth
14
arrowBoth
15
arrowBoth

Question

202502015 202502015

17-05-2024
Mathematics

Answered

Right triangle ABC is represented with right angle at vertex B, towards left. Point D lies on side AC. Segment BD is drawn. Angle BDC is marked as right angle.

The figure shows ∆ABC with m∠ABC = 90°.

The first 10 steps to prove that AB2 + BC2 = AC2 are given in the table. Match the remaining steps to their correct sequence in the proof.

Statement Reason
1. Draw
. construction
2. ∠ABC ≅ ∠BDC Angles with the same measure are congruent.
4. ∠BCA ≅ ∠DCB Reflexive Property of Congruence
4.
AA criterion for similarity
5.
Corresponding sides of similar triangles are proportional.
6. BC2 = AC × DC cross multiplication
7. ∠ABC ≅ ∠ADB Angles with the same measure are congruent.
8. ∠BAC ≅ ∠DAB Reflexive Property of Congruence
9.
AA criterion for similarity
10.
Corresponding sides of similar triangles are proportional.
11.
12.
13.
14.
15.
AB2 + BC2 = AC2
Reason: multiplication
AB2 + BC2 = AC × AC
Reason: segment addition
AB2 = AC × AD
Reason: cross multiplication
AB2 + BC2 = AC(AD + DC)
Reason: Distributive Property
AB2 + BC2 = AC × AD + AC × DC
Reason: addition
11
arrowBoth
12
arrowBoth
13
arrowBoth
14
arrowBoth
15
arrowBoth

Answer :

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lexie2fine lexie2fine · Answer 1 · 2024-05-17T09:14:29-04:00

Answer: AB2 = AC × AD Reason: cross multiplication

BC2 = AC × DC Reason: cross multiplication

AB2 + BC2 = AC × AD + AC × DC Reason: addition

AB2 + BC2 = AC(AD + DC) Reason: Distributive Property

AB2 + BC2 = AC2 Reason: segment addition

The completed proof is:

Draw ∆ABD ∼ ∆ABC (construction)

∠ABC ≅ ∠BDC (Angles with the same measure are congruent.)

∠BCA ≅ ∠DCB (Reflexive Property of Congruence)

∆ABC ∼ ∆DBC (AA criterion for similarity)

Corresponding sides of similar triangles are proportional.

BC2 = AC × DC (cross multiplication)

∠ABC ≅ ∠ADB (Angles with the same measure are congruent.)

∠BAC ≅ ∠DAB (Reflexive Property of Congruence)

∆ABD ∼ ∆ABC (AA criterion for similarity)

Corresponding sides of similar triangles are proportional.

AB2 = AC × AD (cross multiplication)

BC2 = AC × DC (cross multiplication)

AB2 + BC2 = AC × AD + AC × DC (addition)

AB2 + BC2 = AC(AD + DC) (Distributive Property)

AB2 + BC2 = AC2 (segment addition)

The final statement, AB2 + BC2 = AC2, is the Pythagorean theorem.

Step-by-step explanation: