Coordinate Geometry and Trigonometry Question?

Question

A (11, 5), B (5,11), C and D are the vertices of a parallelogram. The points P(17, 8) and Q(21,16) lie on AD and CD respectively. 

i)find the equation of AD and of CD 
ii)find the ratio DQ:QC 
iii)Show that triangle PDQ is isosceles
and also
A right angle triangle shows that AB=hm (adj) and BC= 6m (opp). The point D lies on BC so that BD = 1m and DC=5m. THe angle of CAD is 45 degrees and the angle BAD is an unknown degree (thater). By using the expansion of tan (thater + 45 degrees) or otherwise, find the possible values of h.

Answer 1

(i)

Find the equation of and of .

Step 1:

Parallelogram has coordinate points and .

The points and lie on and .

Consider the coordinate points .

Formula for .

Substitute and in the slope formula.

General form of line equation is .

Substitute and in the general form.

The line passing through the is .

Step 2:

is lies on point of .

Substitute and in the slope formula.

Substitute and in the general form.

The line passing through the is .

Step 3:

is lies on point of .

Here parallel to .

Slope of is .

Substitute and in the general form.

The line passing through the is .

Solution :

line equation is .

line equation is .

Answer 2

(ii)

Find the ratio .

Step 1:

The points and lie on and .

Find .

Here and intersect lines.

The equations are and .

Solve the equations to find point .

Divide each side by .

Substitute in .

The point is .

Step 2:

From (i), parallel to .

Slope of is .

Substitute and in the general form.

The line passing through the is .

Find .

Here and intersect lines.

Solve both equations to find point .

Divide each side by .

Substitute in .

The point is .

Substitute and in the ratio of .

Solution :

The ratio of is .

Answer 3

(iii)

Find is isosceles or not.

Step 1:

The Parallelogram has coordinate points and .

The points are and .

If the is an isosceles triangle, then .

Substitute and in the formula.

Substitute and in the formula.

Since, then is isosceles.

Solution :

then is isosceles.