Question

1. look at the figure. find bc.

Explanation:

Step1: Recall rhombus properties

The diagonals of a rhombus are perpendicular bisectors of each other. Let the intersection of the diagonals be point O. Given half - length of one diagonal $BO = 8$ and side length $AB=15$.

Step2: Apply Pythagorean theorem in right - triangle ABO

In right - triangle $ABO$, by the Pythagorean theorem $AO=\sqrt{AB^{2}-BO^{2}}$. Substitute $AB = 15$ and $BO = 8$ into the formula: $AO=\sqrt{15^{2}-8^{2}}=\sqrt{225 - 64}=\sqrt{161}$.

Step3: Consider congruency of triangles

Since the diagonals of a rhombus are perpendicular bisectors of each other, triangles $ABO$ and $CBO$ are congruent. In right - triangle $BOC$, $BO = 8$ and assume the other leg (part of the other diagonal) is the same as the corresponding part in the upper half of the rhombus. Using the Pythagorean theorem in right - triangle $BOC$ with $BO = 8$ and if we consider the symmetry of the rhombus, and assume the side - length relationship. Since the diagonals of a rhombus bisect each other at right - angles, and we know that in right - triangle $BOC$, if we assume the side - length relationship based on the given information and the property of the rhombus, we can also note that the figure is symmetric about the diagonals. If we consider the fact that the side of the rhombus is equal in length all around. In right - triangle $BOC$, using the Pythagorean theorem $BC=\sqrt{BO^{2}+OC^{2}}$. Since the figure is symmetric and we know from the upper half (using $AB = 15$ and $BO = 8$), and because of the properties of the rhombus, $BC = 15$.

Answer:

15