ABCD is a quadrilateral with angle ABC a right angle. The point D lies on the perpendicular bisector of AB. The coordinates of A and B are (7, 2) and (2, 5) respectively. The equation of line AD is y = 4x − 26. find the area of quadrilateral ABCD

There are several ways to do this:

1. If you join AC, you have a right-angled triangle ABC. Find the length of AB and BC and use
area = (1/2)base x height for its area.
Find angle D by using the slopes of AD and CD,
find the lengths of CD and AD , then area of triangle ACD = (1/2)(CD)(AD)sin D

2. You could use Heron's formula to find the area of triangle ACD, add on the area of the right-angled triangle

3. The simplest and quickest way to find the area of any convex polygon is this:
list the coordinates of the quadrilateral clockwise starting with any point in a column.
repeat the first point you started with.
Area = (1/2)(sum of the diagonal downproducts - sum of the diagonal upproducts)

for yours:
7 2
66/7 82/7
8 15
2 5
7 2

Area = (1/2)[ (7(82/7) + 15(66/7) + 40 + 4) - (2(66/7) + 8(82/7) + 30 + 35)
= (1/2)[1872/7 - 1243/7]
= (1/2)(629/7)
= 629/14

The last method works for any polygon.
Listing the points in a clockwise rotation will result in the negative of the above, so just take the absolute value if you ignore the rotation.
The important thing is to list the points in sequence, and to repeat the first one you started with.

Equation of the perpendicular bisector of line segment AB: y=5x/3 - 4

Coordinates of point D = (66/7, 82/7)
Gradient of BC = 5/3
Coordinates of point C = (8,15)
These are the right answers but how to find the area of quadrilateral? Apparently the answer is 629/14 square units

For the last method, I meant to say :

" list the coordinates of the quadrilateral counter-clockwise starting with any point in a column. "

I am not sure how you got 40+4 and 30+35

btw thx for your help

never mind i got it- it's just like matching

Well, to find the area of a quadrilateral, we can break it down into two triangles. Since ABCD has a right angle at B, triangle ABD and triangle BCD are both right triangles.

Now, to find the length of side AB, we can use the distance formula: √((x2 - x1)^2 + (y2 - y1)^2). Plugging in the coordinates of A and B, we get √((2 - 7)^2 + (5 - 2)^2) = √((-5)^2 + (3)^2) = √(25 + 9) = √34.

Since D lies on the perpendicular bisector of AB, it is equidistant from A and B. This means that the distance from D to both A and B is half of √34, which is √34 / 2.

Now, let's find the height of the quadrilateral. We already know the equation of line AD is y = 4x - 26. To find the y-coordinate of D, we can plug in the x-coordinate of D into this equation. Since D lies on the perpendicular bisector of AB, the x-coordinate of D is the average of the x-coordinates of A and B, which is (7 + 2) / 2 = 4.5. Plugging in 4.5, we get y = 4(4.5) - 26 = 18 - 26 = -8.

So, the height of the quadrilateral is the distance from D to line AB, which is |-8 - 2| = 10.

To find the area of triangle ABD, we can use the formula: area = 1/2 * base * height. The base is √34 and the height is 10, so the area of triangle ABD is 1/2 * √34 * 10 = 5√34.

Similarly, the area of triangle BCD is also 5√34.

Finally, to find the area of quadrilateral ABCD, we just add the areas of the two triangles: 5√34 + 5√34 = 10√34.

Therefore, the area of quadrilateral ABCD is 10√34.

To find the area of quadrilateral ABCD, we need to calculate the area of triangle ABC and triangle ACD separately, and then add them together.

First, let's find the length of AB using the distance formula:

AB = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((2 - 7)² + (5 - 2)²)
= √((-5)² + 3²)
= √(25 + 9)
= √34

Since D lies on the perpendicular bisector of AB, it means that AD is equal to BD. So, the length of AD is also √34.

Now, let's find the height of triangle ABC. Since angle ABC is a right angle, the height is the perpendicular distance from point C to line AB. We can find this height by determining the perpendicular distance from point C to line AD, which is the same as the perpendicular distance from point C to line BD.

The equation of line AD is y = 4x - 26. To find the perpendicular distance from point C (7, 2) to this line, we can use the formula:

d = |Ax + By + C| / √(A² + B²)

Where A, B, and C are the coefficients of the equation Ax + By + C = 0, and (x, y) are the coordinates of the point.

In this case, the equation of line AD can be rewritten as 4x - y - 26 = 0. So, A = 4, B = -1, and C = -26.

Plugging in the values, we get:

d = |(4 * 7) + (-1 * 2) + (-26)| / √((4)² + (-1)²)
= |28 - 2 - 26| / √(16 + 1)
= 0 / √17
= 0

So, the height of triangle ABC is 0, which means the area of triangle ABC is 0 as well.

The area of triangle ACD can be calculated as half the product of the base AD (√34) and the height (0), which is also 0.

Therefore, the area of quadrilateral ABCD is the sum of the areas of triangle ABC and triangle ACD, which is 0 + 0 = 0.