Respuesta :
The Pythagorean Theorem tells us that in a right angle, the sum of the squares lengths of the legs of the triangle equal the squared length of the hypotenuse.
By angle similarity of triangles we know that if we have three triangles △ADB △ABC △BDC triangles are similar then the sides of the triangles are proportional. AD:AB=AB:AC=AC::BC= BC:DC
AD/AB=AB/AC AC/BC=BC/DC
Now we cross multiply
AD*AC=AB*ABAC*DC=BC*BC
This is the same as
AD*AC=AB2AC*DC=BC2
AD*AC+AC*DC=AB2+BC2
Distributive property
AC(AD+DC)=AB2+BC2
But by construction
AD+DC=AC
Substituting
AC*AC=AB2+BC2
So we have
AB2+BC2=AC2
This proves that the sum of the squares of the lengths of the legs of the triangle equals the squared length of the hypotenuse.
By angle similarity of triangles we know that if we have three triangles △ADB △ABC △BDC triangles are similar then the sides of the triangles are proportional. AD:AB=AB:AC=AC::BC= BC:DC
AD/AB=AB/AC AC/BC=BC/DC
Now we cross multiply
AD*AC=AB*ABAC*DC=BC*BC
This is the same as
AD*AC=AB2AC*DC=BC2
AD*AC+AC*DC=AB2+BC2
Distributive property
AC(AD+DC)=AB2+BC2
But by construction
AD+DC=AC
Substituting
AC*AC=AB2+BC2
So we have
AB2+BC2=AC2
This proves that the sum of the squares of the lengths of the legs of the triangle equals the squared length of the hypotenuse.
Answer:
To prove the Converse of Pythagorean Theorem:
Step-by-step explanation:
1. Draw a right triangle (Triangle ABC)
2. Draw an altitude from and A to the hypotenuse creating 2 new triangles
3. The part that intersects the hypotenuse will be point D, this makes a right angle
Triangle DBA and triangle ABC both have right angles and share angle B, this means that triangle ABC and triangle DBA are congruent by the AA postulate. Since triangle DAC and triangle ABC both have right angles(angle D and angle A) and they share angle C, triangle DAC and ABC are congruent by the AA postulate as well.
The ratios can be set up as:
AC/BC = DC/AC & AB/BC = DB/BA
Once cross multiplied, the result should be:
(AC)² = (BC)(DC) & (AB)² = (BC)(DB)
Now add both sides together to get:
(AC)² + (AB)² = BC(DC) + (BC)(DB)
To make things easier, factor out the like terms:
(AC)² + (AB)² = BC(DC + DB)
When looking at the triangle(must draw to see), it shows that (DC + DB) is actually side BC. This leaves:
(AC)² + (AB)² = BC(BC) → (AC)² + (AB)² = (BC)²
The Pythagorean theorem says that a² + b² = c². Knowing this, on the original triangle, side AB = a, side AC = b, and side BC = c. Knowing this:
(AC)² + (AB)² = (BC)² → a² + b² = c²
This proves the Converse of the Pythagorean Theorem.
