Question

Which is the last sentence of the proof?

Because f + e = 1, a2 + b2 = c2.
Because f + e = c, a2 + b2 = c2.
Because a2 + b2 = c2, f + e = c.
Because a2 + b2 = c2, f + e = 1.


Because △ABC and △CBD both have a right angle, and the same angle B is in both triangles, the triangles must be similar by AA. Likewise, △ABC and △ACD both have a right angle, and the same angle A is in both triangles, so they also must be similar by AA. The proportions and are true because they are ratios of corresponding parts of similar triangles. The two proportions can be rewritten as a2 = cf and b2 = ce. Adding b2 to both sides of first equation, a2 = cf, results in the equation a2 + b2 = cf + b2. Because b2 and ce are equal, ce can be substituted into the right side of the equation for b2, resulting in the equation a2 + b2 = cf + ce. Applying the converse of the distributive property results in the equation a2 + b2 = c(f + e).

Answer 1

B. Because f + e = c,
`a^(2)` +
`b^(2)`=
`c^(2)`.

Explanation:

EDGE2021

Answer 2

Because f+e=c,
`a^2+b^2=c^2`.

Explanation:

Given
`\triangle ABC` and
`\triangle CBD` are right triangles.

To prove that
`a^2+b^2=c^2`

Because
`\triangle ABC\; and \;\triangle CBD` both have a right angle. It means one angle of both triangle is of
`90^(\circ)`.

And tha angle B is same in both triangles .Therefore, triangles must be similar by AA similarity.

Similarly ,
`\triangle ABC` and
`\triangle  CBD` are both right triangles and angle A is common in both triangles . So , they are similar by AA similarity .

Similarity: When two triangles are simialr then their corresponding angles are equal and their corresponding sides are in equal proportion.

Therefore , the two proportions can be rewritten as

`a^2=cf` ( I equation )

`b^2=ce` ( II equation )

Adding
`b^2` on both side of I equation then we can write equation I as

`a^2+b^2=b^2+cf`

`a^2+b^2= ce+cf`

Because
`b^2` and ce are equal and substitute on the right side of equation I

Applying the converse of distributive property in above eqaution then we get new equation

`a^2+b^2= c( f+e)`

Because distributive property a.(b+c)=a.b+a.c

`a^2+b^2=c^2`

Because
`e+f = c^2`.