Question

which statement is necessarily true if \\( \overline { b d } \\) is an altitude to the hypotenuse of right \\( \delta a b c \\) ?
a. \\( \delta a d b \cong \delta b d c \\)
b. \\( \delta a d b \sim \delta b d c \\)
c. \\( \frac { a b } { b c } = \frac { a c } { b d } \\)
d. \\( \angle b a c \cong \angle b d c \\)

Explanation:

Step1: Recall the properties of similar triangles

When an altitude is drawn to the hypotenuse of a right triangle, the two smaller triangles formed are similar to the original triangle and to each other.
In right triangle \(ABC\) with altitude \(BD\) to the hypotenuse \(AC\), \(\triangle ADB\) and \(\triangle BDC\) are right triangles.
For \(\triangle ADB\) and \(\triangle BDC\):

• \(\angle ADB=\angle BDC = 90^{\circ}\)
• \(\angle ABD+\angle CBD = 90^{\circ}\) and \(\angle C+\angle CBD=90^{\circ}\), so \(\angle ABD=\angle C\) (by the property that if two angles are complementary to the same angle, they are equal)

Step2: Use the AA (Angle - Angle) similarity criterion

The AA similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.
In \(\triangle ADB\) and \(\triangle BDC\), \(\angle ADB=\angle BDC\) (both are right angles) and \(\angle ABD=\angle C\). So, \(\triangle ADB\sim\triangle BDC\) by the AA similarity criterion.

Now check other options:

• Option A: \(\triangle ADB\cong\triangle BDC\) is not necessarily true. Congruence requires equal side lengths in addition to equal angles. There is no information given about side - length equality.
• Option C: \(\frac{AB}{BC}=\frac{AC}{BD}\) is not a valid proportion from the similarity of the triangles formed by the altitude to the hypotenuse of a right - triangle. The correct proportions are \(AB^{2}=AD\times AC\), \(BC^{2}=CD\times AC\) and \(BD^{2}=AD\times DC\) from the geometric mean theorem.
• Option D: \(\angle BAC\cong\angle BDC\). \(\angle BAC\) is an acute angle of the original right - triangle \(ABC\), and \(\angle BDC = 90^{\circ}\). So, they are not congruent.

Answer:

B. \(\triangle ADB\sim\triangle BDC\)