Question

select all the correct answers.
in $\triangle abc$, $\overline{bd}$ is perpendicular to $\overline{ac}$ as shown in the figure.
which equations are true?
$ac^2 + cb^2 = ab^2$
$ab^2 + bd^2 = ad^2$
$bc^2 + cd^2 = bd^2$
$ab^2 + bc^2 = ac^2$
$ad^2 + db^2 = ab^2$
$bd^2 + dc^2 = bc^2$

Explanation:

Step1: Identify right triangles

We have 3 right triangles: $\triangle ABD$ (right at $D$), $\triangle CBD$ (right at $D$), $\triangle ABC$ (right at $B$).

Step2: Apply Pythagoras to $\triangle ABD$

For right $\triangle ABD$, hypotenuse is $AB$.
$AD^2 + DB^2 = AB^2$

Step3: Apply Pythagoras to $\triangle CBD$

For right $\triangle CBD$, hypotenuse is $BC$.
$BD^2 + DC^2 = BC^2$

Step4: Apply Pythagoras to $\triangle ABC$

For right $\triangle ABC$, hypotenuse is $AC$.
$AB^2 + BC^2 = AC^2$

Step5: Eliminate incorrect options

Options that reverse the hypotenuse and legs are false.

Answer:

• $AB^2 + BC^2 = AC^2$
• $AD^2 + DB^2 = AB^2$
• $BD^2 + DC^2 = BC^2$