ABCD is a parallelogram with sides AB=12cm,BC=10cm and diagonal Ac=16cm.find the area of parallelogram. Also find the distance between its shorter side.
several ways of doing this:
1. use Heron's formula
split the parallelogram into 2 congruent triangles, then apply using
http://www.mathwarehouse.com/geometry/triangles/area/herons-formula-triangle-area.php
2. Use the cosine law to find the obtuse angle between the two sides 12 and 10
16^2 = 12^2 + 10^2 - 2(12)(10)cosØ
once you have angle Ø
area of ||gram = (12)(10)sinØ
Once you have the area,
use 10 as the base, and let the distance between the two short sides by h, your required altitude.
area = base x height
area = 10h
solve for h, since you have the area from above
To find the area of a parallelogram, you can use the formula A = base * height, where the base is any one of the sides and the height is the perpendicular distance between the base and the opposite side.
In this case, sides AB and CD are parallel and have the same length of 12 cm. Let's take AB as the base. As shown in the diagram below:
B ________ C
\ /
\ /
\ /
\ /
\/
A
The perpendicular distance between AB and the opposite side CD is the length of the segment CE.
First, we need to find the length of CE. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In triangle ACE, we have AC as the hypotenuse, with a length of 16 cm, and AE as the unknown side.
Using the Pythagorean theorem:
AC^2 = AE^2 + CE^2
16^2 = AE^2 + CE^2
256 = AE^2 + CE^2
We know that AE is the length of side AB, which is 12 cm. Substituting this into the equation:
256 = 12^2 + CE^2
256 = 144 + CE^2
CE^2 = 256 - 144
CE^2 = 112
CE ≈ √112
CE ≈ 10.59 cm (rounded to two decimal places)
Now that we know the length of CE, we can calculate the area of the parallelogram using the formula A = base * height:
A = AB * CE
A = 12 cm * 10.59 cm
A ≈ 127.08 cm² (rounded to two decimal places)
Therefore, the area of the parallelogram is approximately 127.08 cm².
To find the distance between the shorter side, which is BC, and the opposite side, we can use the distance formula, which calculates the distance between two points in a coordinate system.
In this case, point B and point C define the shorter side BC. Let's represent the coordinates of point B as (0, 0) and the coordinates of point C as (10, 0), since BC has a length of 10 cm.
Using the distance formula:
Distance = √(x2 - x1)^2 + (y2 - y1)^2
Let's consider point D as (x, y), where x is the unknown x-coordinate and y is the unknown y-coordinate.
The distance between BC and the opposite side, which is AD, can be calculated as the distance between points (x, y) and (10, 0).
Using the distance formula for points (x, y) and (10, 0):
Distance = √(10 - x)^2 + (0 - y)^2
We know that AD is the hypotenuse of the right triangle BCE, and we've already found CE, which is approximately 10.59 cm.
The triangle BCE is similar to triangle ABD, so the ratio of their corresponding sides is the same.
Using proportions:
CE / BC = AD / AB
10.59 cm / 10 cm = AD / 12 cm
(10.59 * 12 cm) / 10 cm = AD
AD ≈ 12.71 cm (rounded to two decimal places)
Therefore, the distance between the shorter side BC and the opposite side AD is approximately 12.71 cm.