Given: ABCD is a rectangle, AC∩BD= O, m∠ABD=30°, AC=16 in. Find BC
AC = BD = 16 (diagonals of rectangle)
m∠BDC=30° because AB || DC
sin 30 = BC / BD = BC / 16
so
BC = 16 sin 30 = 16 * 1/2
To find the length of BC, we can use the properties of a rectangle and the given information.
1. Since ABCD is a rectangle, opposite sides are congruent. This means that AB = CD and BC = AD.
2. We are given that AC = 16 inches. Since AC is the diagonal of the rectangle, we can use the Pythagorean theorem to find the length of AB or CD.
Using Pythagorean theorem: AC^2 = AB^2 + BC^2
Substitute the known values: (16)^2 = AB^2 + BC^2
3. To solve for BC, we need to find the value of AB. Given that m∠ABD = 30°, we can use trigonometric ratios to find AB.
In a right triangle ABD, the angle ABD is 30° and the side opposite to it is AB.
Using trigonometry: sin(30°) = AB/BD
Substitute the known values: sin(30°) = AB/BD
4. BD is a side of the rectangle, so it is equal to BC. Substitute BC for BD in the equation from step 3.
5. Solve equation (2) for AB and substitute the value of AB in equation (1).
Solve for AB in equation (2): sin(30°) = AB/BC
Rearrange the equation: AB = BC * sin(30°)
6. Substitute the value of AB from step 5 in equation (1) and solve for BC.
(16)^2 = (BC * sin(30°))^2 + BC^2
Simplify: 256 = BC^2 * sin^2(30°) + BC^2
Factor out BC^2: 256 = BC^2 * (sin^2(30°) + 1)
Simplify: 256 = BC^2 * (1/4 + 1)
Combine fractions: 256 = BC^2 * (5/4)
Divide both sides by (5/4): BC^2 = 256 / (5/4)
Simplify: BC^2 = 256 * (4/5)
BC^2 = 204.8
Taking the square root of both sides: BC ≈ √204.8
BC ≈ 14.3 inches
Therefore, the length of BC is approximately 14.3 inches.