Question

kayla wants to prove the pythagorean theorem: if a triangle is a right triangle, then the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

triangle abc with right angle at b

select the appropriate rephrased statement for kaylas proof.

choose 1 answer:

a) in △abc, if ∠b = 90°, then ab² = bc² + ac².

b) in △abc, if ab² = bc² + ac², then ∠b = 90°.

c) in △abc, if ∠b = 90°, then ab² + bc² = ac².

d) in △abc, if ab² + bc² = ac², then ∠b = 90°.

Explanation:

The Pythagorean theorem states that in a right - triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In \(\triangle ABC\), \(\angle B = 90^{\circ}\), so the hypotenuse is \(AC\) and the other two sides are \(AB\) and \(BC\). So the correct rephrased statement should be: If \(\angle B=90^{\circ}\) (the triangle is right - angled at \(B\)), then \(AB^{2}+BC^{2}=AC^{2}\).

• Option A: It has the wrong relationship between the sides. It says \(AB^{2}=BC^{2}+AC^{2}\), which is incorrect.
• Option B: This is the converse of the Pythagorean theorem (it states that if the square of one side is equal to the sum of the squares of the other two sides, then the triangle is right - angled), not the Pythagorean theorem itself.
• Option C: This is the correct rephrasing of the Pythagorean theorem for \(\triangle ABC\) with \(\angle B = 90^{\circ}\).
• Option D: This is also the converse of the Pythagorean theorem.

Answer:

C. In \(\triangle ABC\), if \(\angle B = 90^{\circ}\), then \(AB^{2}+BC^{2}=AC^{2}\)