Find The Area of A Quadrilateral

[LLS]

The Area of A Quadrilateral Given A Square

Homework Statement

In a square PQRS, point T is the midpoint of side QR. If the area of square PQRS is 3, what is the area of quadrilateral PQTS?

Homework Equations

area = 1/2 base * height

The Attempt at a Solution

Side QR = 1.737

Side RS = 1.737

TR = .8685

area = 1/2(1.737) * .8685 = .25

The area of the triangle = .25

3 - .25 = 2.75

The area of quadrilateral PQTS = 2.75

Is my answer right?

 

[dynamicsolo]

area = 1/2(1.737) * .8685 = .25

Check your arithmetic here: you have one-half of a number close to 1.8, which you multiplied by a number close to 0.9, and got 0.25...(?)

I have another suggestion. Draw a picture of this square with the line segment PT added. What is the area of triangle PQT? (Incidentally, because of the symmetry of the geometry here, you don't even need to use the formula for the area of a triangle.)

Now, the quadrilateral PQTS is made up of half the square plus a triangle of the same area as PQT. So what would this quadrilateral's area be?

 

[Dr Zoidburg]

Sometimes it's easier to stick with whole numbers and fractions:
length of side = \sqrt{3}
length of QT = \frac{1}{2}\sqrt{3}

area of triangle QTS = \frac{1}{2}*\frac{\sqrt{3}}{2}*\sqrt{3}
=\frac{\sqrt{3}}{2}*\frac{\sqrt{3}}{2}
and what does \sqrt{3}*\sqrt{3} = ?
divide that by 4 (1/2 * 1/2) for your answer.

Then take that away from 3.

 

[LLS]

Sometimes it's easier to stick with whole numbers and fractions:
length of side = \sqrt{3}
length of QT = \frac{1}{2}\sqrt{3}

area of triangle QTS = \frac{1}{2}*\frac{\sqrt{3}}{2}*\sqrt{3}
=\frac{\sqrt{3}}{2}*\frac{\sqrt{3}}{2}
and what does \sqrt{3}*\sqrt{3} = ?
divide that by 4 (1/2 * 1/2) for your answer.

Then take that away from 3.

√3*√3 = 3

3 - 3/4 = 2.25

The quad = 2.25?

I don't think that answer is correct.

 

[LLS]

I may have mad a mistake in the math. Is the answer 2.75?

 

[dynamicsolo]

The triangle you are describing has one-quarter the area of the square. (Take the midpoint on the side opposite QR, which is PS. A line straight from T to that other midpoint divides the square in two. The line from P to T divides that rectangle in half diagonally, so triangle PQT has one-quarter of 3 units or 0.75.

The quadrilateral PQTS is made up of the upper half of the square plus a triangle with the same area as PQT. So it has three-quarters of the area of the whole square or
(3/4) · 3 = 2.25 units.

 

[LLS]

The triangle you are describing has one-quarter the area of the square. (Take the midpoint on the side opposite QR, which is PS. A line straight from T to that other midpoint divides the square in two. The line from P to T divides that rectangle in half diagonally, so triangle PQT has one-quarter of 3 units or 0.75.

The quadrilateral PQTS is made up of the upper half of the square plus a triangle with the same area as PQT. So it has three-quarters of the area of the whole square or
(3/4) · 3 = 2.25 units.

Thank you