Question

Justin is using the figure shown below to prove Pythagorean Theorem using triangle similarity: In the given triangle ABC, angle A is 90o and segment AD is perpendicular to segment BC. The figure shows triangle ABC with right angle at A and segment AD. Point D is on side BC. Which of these could be a step to prove that BC2 = BA2 + CA2?

Answer 1

Option fourth "By the addition property of equality, AC² plus AB² = BC multiplied by DC plus AB² is correct.

What is the Pythagoras theorem?

The square of the hypotenuse in a right-angled triangle is equal to the sum of the squares of the other two sides.

The figure is missing.

The right-angle triangle is shown in the picture; please refer to the picture.

The missing options are attached; please refer to the picture.

We have a right-angle triangle shown in the picture.

The larger triangle ABC is similar to the smaller triangles as follows.

ΔABC ~ ΔDBA     (By AA similarity)

ΔABC ~ ΔDAC   (By AA similarity)

From the above similarity:

AC² or AB²

$\rm \dfrac{AB}{BD} = \dfrac{BC}{AB}$

After cross multiplication:

AB² = BC×BD

Now,

$\rm \dfrac{AC}{CD} = \dfrac{BC}{AC}$

AC² = BC×BD

AB² + AC² =  BC×BD + BC×BD

AB² + AC² =  2BC×BD

Thus, the option fourth "By the addition property of equality, AC² plus AB² = BC multiplied by DC plus AB² is correct.

Learn more about Pythagoras' theorem here:

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