Math Snap

STEP 1

1. Triangle $ABC$ is a right triangle with $\angle B = 90^\circ$.
2. $\overline{BD}$ is the altitude from $B$ to $\overline{AC}$.
3. The length of $\overline{AB}$ is $15$.
4. The length of $\overline{AC}$ is $23$ greater than the length of $\overline{AB}$.

STEP 2

1. Determine the length of $\overline{AC}$.
2. Use the Pythagorean theorem to find the length of $\overline{BC}$.
3. Use the geometric mean theorem to find the length of $\overline{BD}$.
4. Calculate $\frac{BC}{BD}$.

STEP 3

Determine the length of $\overline{AC}$:
Since $\overline{AC}$ is $23$ greater than $\overline{AB}$, we have:
$$AC = AB + 23 = 15 + 23 = 38$$

STEP 4

Use the Pythagorean theorem to find $\overline{BC}$:
In $\triangle ABC$, since $\angle B = 90^\circ$, we have:
$$AC^2 = AB^2 + BC^2$$ Substitute the known values:
$$38^2 = 15^2 + BC^2$$ $$1444 = 225 + BC^2$$ $$BC^2 = 1444 - 225$$ $$BC^2 = 1219$$ $$BC = \sqrt{1219}$$

STEP 5

Use the geometric mean theorem to find $\overline{BD}$:
The geometric mean theorem states that in a right triangle, the altitude to the hypotenuse is the geometric mean of the segments it divides the hypotenuse into. Let $AD = x$ and $DC = y$. Then:
$$BD^2 = AD \cdot DC$$ Since $AD + DC = AC = 38$, and using the properties of the right triangle, we have:
$$BD = \frac{AB \cdot BC}{AC}$$ Substitute the known values:
$$BD = \frac{15 \cdot \sqrt{1219}}{38}$$

Calculate $\frac{BC}{BD}$:
$$\frac{BC}{BD} = \frac{\sqrt{1219}}{\frac{15 \cdot \sqrt{1219}}{38}}$$ Simplify the expression:
$$\frac{BC}{BD} = \frac{38}{15}$$ The value of $\frac{BC}{BD}$ is:
$$\boxed{\frac{38}{15}}$$